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% Files
% weyl_2.c  scattering code
%file_in.dat input file to weyl_2.c
% complex_exact.c exact code
% ex2c.in - runs on gelfand
% daub.c code to fields
% daub_3d.py
% daub_r.gp  generate 3d plots
% daub_i.gp  generate 3d plots
% daub_r0.gp generate 3d plots
% daub_i0.gp generate 3d plots
% gamma_coeff.py numerical computation of gamma coeff
% pyth calls python codes - prints figures from weyl_2
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\begin{document}

\title{Path integrals and the discrete Weyl representation} %Title of paper

\author{W.~N.~Polyzou}
\email{polyzou@uiowa.edu}
\thanks{This work supported by the U.S. Department of Energy,
  Office of Science, Grant \#DE-SC16457}
\affiliation{Department of Physics and Astronomy\\
  The University of Iowa\\
  Iowa City, IA 52242, USA}

\date{\today}

\begin{abstract}

This paper uses Schwinger's representation of quantum systems of a
finite number of degrees of freedom to formulate discrete path
integral representations of the dynamics. The representation starts
with an observable with a finite number of distinct eigenvalues.
Using that observable Schwinger constructs complementary pairs of
finite dimensional unitary operators that are finite dimensional
analogs of the irreducible Weyl algebra.  These can be decomposed into
products of irreducible sub-algebras that reduce to q-bit gates for
the case of $2^N$ degrees of freedom.  In the limit of a large number
of discrete degrees of freedom these representations can be used to
model quantum systems with continuous degrees of freedom.  This limit
recovers the continuum Weyl algebra.  Path integrals can be formulated
in the discrete case.  They have the advantage that the number of
paths for a finite number of time slices is finite.  In this work the
path integral is interpreted as the expectation value of a potential
functional with respect to a complex probability distribution on the
space of ``paths''.  An application to scattering from a short-range
potential is given.  Multi-resolution wavelet bases are used exactly
represent a local $\phi^4(x)$ quantum field theory as a theory with an
infinite number of discrete local modes.  The discrete path integral
is illustrated by computing time evolution when this theory is
truncated to two coupled modes.
  
\end{abstract}

\maketitle 

\bigskip

\section{Introduction}\label{intro}

Quantum computers consist of a finite number, $M$, of two state
systems.  The resulting Hilbert space for this quantum system is
finite dimensional.  Elementary gates are used to build an irreducible
algebra of operators on this space that can be used to model complex
systems.  The goal is to time evolve these systems to solve problems
that are difficult to solve classically.  This requires time evolving
vectors with large numbers of components and measuring the results.
At the computational level results are achieved using finite quantum
systems.  Julian Schwinger's textbook \cite{schwinger} treatment of
measurement theory generalizes the standard measurement theoretic
treatment of the Stern Gerlach experiment to a system of a finite
number of degrees of freedom.  It is a natural framework for numerical
treatments of path integrals, although these applications are limited
on classical computers.  Schwinger begins with a quantum observable
that has a finite number of possible outcomes.  He then constructs a
complementary set of unitary operators - one that has the same
eigenvectors as the original observable and one that cyclically shifts
the eigenvectors.  These two unitary operators are finite degree of
freedom analogs of the Weyl algebra, which is the exponential form of
the canonical commutation relations.  In this case there is no
identification of these observables with coordinates or momenta.  When
$M$ can be expressed as a product of prime factors, he shows that this
algebra can be decomposed into products of irreducible sub-algebras
acting on independent sets of prime numbers of degrees of freedom.
When $M=2^L$ these elementary unitary operators can be represented by
the q-bit gates $\sigma_x$ and $\sigma_z$.

While Schwinger's representation provides a general structure theorem
for quantum systems of a finite numbers of degrees of freedom, there
are natural limits that provide models of quantum systems based on
commuting observables with continuous eigenvalues.  These systems can
be approximated by finite systems with large number of degrees of
freedom.
%When $M=2^L$ the irreducible building blocks of the system
%are q-bits.
The discrete representation leads to a discrete formulation of the
path integral where for $N$ time steps there are a finite number, $M^N$,
``paths'' that pass through the $M$ allowed values of one of the
observables at each time step.  For a canonical system that is
quadratic in the ``momentum'' variables, the amplitude for free
propagation between time steps defines a ``complex probability'' on
the space of paths.  The path integral can be interpreted as the small
time step limit of the expectation value of an interaction functional
of the discrete of paths between time steps.  An application of this
discrete path integral to scattering in one dimension is illustrated
in section \ref{scatt}.  The discrete path integral is used to
approximate sharp-momentum half-shell transition matrix elements.  The
same method can be applied to time evolve quantum fields using an
exact discrete multi-resolution representation of the field
algebra. The computation of time evolution of a volume and resolution
truncated quantum field theory using the discrete path integral is
discussed in section \ref{qft}.  It is illustrated using a trivial two
mode truncation of the theory.

While the Schwinger representation does not solve any of the problems
that quantum computers are designed to solve, it leads to a simple
framework for modeling general quantum problems as finite quantum
systems, where these finite systems can also be represented by products of
elementary 2 or 3 state quantum systems.  The 2-3 state algebras are
more localized and should provide a more practical representation for
quantum computations.  In general applications are limited by the
dimension of the vectors that represent realistic systems.

The next section provides a brief discussion of the role of Hilbert
spaces in the formulation of three valued quantum logic represented by
q-bits.  This emphasizes the relevant difference between digital and
quantum computing.  Section \ref{weyl} provides a summary of
Schwinger's construction of irreducible algebras of systems of $M$
degrees of freedom.  Section \ref{qbit} discusses the factorization of
a $2^M$ degree of freedom system into a direct product of irreducible
two-degree of freedom systems which results in a representation of the
algebra of section \ref{weyl} in terms of qbits.  Section \ref{cont}
discusses the limit to a system with continuous eigenvalues.  In this
case the general construction requires additional boundary conditions.
Section \ref{complex} gives a short discussion of the subject of
complex probabilities, which will be used in the discrete formulation
of the path integral.  Section \ref{path} discusses the treatment of
time evolution using a discrete formulation of the path integral as
the expectation value of a potential functional with respect to a
complex probability on a finite sample space of paths.  Section
\ref{scatt} illustrates the application of the path integral in
section \ref{path} to scattering in one dimension.  Section \ref{qft}
discusses an exact discrete multi-resolution representation of a
scalar field theory and uses the path integral in section \ref{path}
to time evolve a two field-mode truncation of the theory.  Section
\ref{sum} gives a summary and conclusion.


\section{Quantum logic}\label{logic}

Classically if a system is prepared in a state $A$ and a later
measurement tests if it will be detected in state $B$, there are two
possible outcomes, true or false.  This leads to a two valued system of
logic that is encoded in the bits used in digital computing.
In quantum mechanics there are three possibilities - the final
system will always be measured to be in the state $B$, it will never
be measured to be in the state $B$, or there is a finite probability
$P$, with $0<P<1$, that it will be measured to be in state $B$. This
leads to a three-valued logic or quantum logic.

The three valued logic of quantum mechanics \cite{birkhoff} has a
straightforward geometrical interpretation.  If state $A$ is
represented by a one-dimensional subspace of a Hilbert space and state
$B$ is represented by another one dimensional subspace then there are
three possibilities - (1) the subspace $B$ is the
subspace $A$, (2) the subspace $B$ is orthogonal to the subspace $A$,
or a non-0 vector in $A$ has a non-zero projection on the subspace
$B$.  

In the quantum case states are represented by vectors or rays, $\vert a
\rangle$ in a Hilbert space.  Quantum probabilities are expressed in
terms of the Hilbert space inner product:
\[
P_{ab} := {\langle a \vert b \rangle \langle b \vert a \rangle \over \langle a \vert a \rangle\langle b \vert b \rangle }
\]
which is independent of the vectors in the rays.
The three possibilities correspond to
\beq
(1) \qquad P_{ab}=1 
\qquad (2) \qquad
P_{ab}=0
\qquad (3) \qquad
0<P_{ab}<1.
\eeq
When the Hilbert space is two dimensional the difference
in these two types of logic is encoded in bits or q-bits respectively.

In quantum mechanics observable quantities are represented by linear
operators $A$ on a Hilbert space.  The only possible outcomes for
measuring $A$ are one of its eigenvalues, $a_n$ (this assumes $A$ is a
normal operator whose eigenvectors form a basis). In this case the
Hilbert space has a decomposition ${\cal H} = \oplus_n {\cal H}_n$,
where the ${\cal H}_n$ are $A$-invariant subspaces of ${\cal H}$.

The mean value of a measurement of $A$ in state $\vert b\rangle$ is  
\beq
\langle b \vert A \vert b \rangle =
\sum_n P_{a_nb} a_n
\qquad a_n \mbox{ eigenvalue of A} 
\eeq
which is the weighted average of the quantum probabilities
for a  measurement of $b$ to be in one of the eigenstates of $A$.

\section{Schwinger's discrete Weyl Algebra}\label{weyl}

This section reviews Schwinger's \cite{schwinger} method of
constructing an irreducible algebra of complementary unitary operators
for quantum systems of a finite number of degrees of freedom.  This
construction generates a finite degree of freedom version of the Weyl
(exponential) form of the canonical commutations relations.  This
algebra can be used to build discrete models of any quantum system.
This construction is essentially the same as the treatment of the
quantum Fourier transform discussed in \cite{nielsen} and elsewhere.
   
Let ${\cal H}$ be a $M$-dimensional complex Hilbert space.  Let $A$ be
a normal operator on ${\cal H}$ with $M$ distinct eigenvalues
and unit normalized eigenvectors:
\beq
A \vert a_m \rangle =a_m \vert a_m \rangle  \qquad m=1,\cdots, M
\qquad a_m \not= a_n \mbox{ for } m\not=n .
\label{s1}
\eeq
Define the operator $U$ on ${\cal H}$ that cyclically shifts the
eigenvectors of $A$:
\beq
U\vert a_m \rangle = \vert a_{m+1} \rangle \qquad m<M
\qquad
U\vert a_M \rangle = \vert a_1 \rangle .
\label{s2}
\eeq
In what follows the labels $m$ on eigenvectors and eigenvalues are treated
as integers mod $M$ so 0 is identified with $M$, $1$ with $M+1$ etc..
$U$ defined by (\ref{s2}) is unitary since
\beq
UU^{\dagger}= \sum_{m=1}^M U\vert a_m \rangle \langle a_m \vert U^{\dagger}=
\sum_{m=1}^{M-1} \vert a_{m+1} \rangle \langle a_{m+1} \vert 
+ \vert a_{1} \rangle \langle a_{1} \vert =
\sum_{m'=1}^M \vert a_{m'} \rangle \langle a_{m'} \vert = I .
\label{s3}
\eeq
Since $M$ applications of $U$ leaves all $M$
basis vectors, $\vert a_m \rangle$,  unchanged, it follows that $U^M=I$.
This
means that the characteristic
polynomial of $U$ is $P(\lambda) = \lambda^M-1 =0$.  The eigenvalues
of $U$ are the $M$ roots of $1$:
\beq
\lambda = u_m = e^{2\pi  m i\over M}.
\label{s4}
\eeq
Let  $\vert u_m \rangle$ denote the associated eigenvectors:
\beq
U \vert u_m \rangle = u_m \vert u_m \rangle
\label{s5}
\eeq
with unit normalization
\beq
\langle u_m \vert u_n \rangle = \delta_{mn}.
\label{s6}
\eeq
The normalization does not fix the phase 
which will be fixed later.  Since both $U^M=I$  and $u_n^M=1$
it follows that 
\[
0=
(U^M-I ) = {1 \over u_n^M}  (U^M-I  )=
\]
\beq
\left (({U\over u_n})^M-I\right ) = \prod_{m=1}^M ({U\over u_n}- {u_m \over u_n}) =
({U\over u_n}-1)(1 +{U\over u_n} + ({U\over u_n})^2 +\cdots + ({U\over u_n})^{M-1}).
\label{s7}
\eeq
Since this expression is identically zero and $({u_m\over u_n}-1)\not=0$
for $m\not= n$ it follows that
\beq
1 +{U\over u_n} + ({U\over u_n})^2 +\cdots ({U\over u_n})^{M-1}
= c \vert u_n \rangle \langle u_n \vert 
\label{s8}
\eeq
for some constant $c$.
Applying (\ref{s8}) to $\vert u_n \rangle$ implies that the constant $c=M$.
This results in an expression for the projection operator on each eigenstate
of $U$ as a degree $M-1$ polynomial in $U$
\beq
\boxed{
\vert u_n \rangle \langle u_n \vert =
{1 \over M} \sum_{m=1}^M ({U\over u_n})^m =
{1 \over M} \sum_{m=0}^{M-1} ({U\over u_n})^m .
}
\label{s9}
\eeq
%In the above and what follows integral lables on the states and
%eigenvectors will be understood to be integers mod $M$.
Using (\ref{s9}) it follows that
\beq
\langle a_k \vert u_n \rangle \langle u_n \vert a_k \rangle  =
{1 \over M} \sum_{m=0}^{M-1} \langle a_k \vert ({U\over u_n})^m \vert a_k \rangle 
=
{1 \over M} \sum_{m=0}^{M-1} ({1\over u_n})^m\langle a_k \vert a_{k+m} \rangle
= {1 \over M} .
\label{s10}
\eeq
This means for any $k$ and $n$ that
\beq
\vert \langle a_k \vert u_n \rangle \vert = {1 \over \sqrt{M}} .
\label{s11}
\eeq
The interpretation is that if the system is prepared in any eigenstate
of $U$ and $A$ is subsequently measured, then the probability of
measuring any of the eigenvalues of $A$ is the same (1/M).  This means that
all of the information about the identity of the initial eigenstate of
$U$ is lost after measuring $A$.  This is the condition for the
observables $A$ and $U$ to be complementary.

The phase of $\vert u_n \rangle$ is defined by choosing
\beq
\langle a_M \vert u_n \rangle = \langle u_n \vert a_M \rangle
= {1 \over \sqrt{M}} .
\label{s12}
\eeq
It then follows from (\ref{s12}) that
\[
\langle a_k \vert u_n \rangle \langle u_n \vert a_M \rangle =
\langle a_k \vert u_n \rangle {1 \over \sqrt{M}} =
\]
\beq
{1 \over M} \langle a_k \vert \sum_{m=1}^M u_n^{-m} \vert a_m \rangle =
{1 \over M} u_n^{-k} = {1 \over M} e^{-2\pi i nk/M}
\label{s13}
\eeq
which gives the inner product
\beq
\langle a_k \vert u_n \rangle = {1 \over \sqrt{M}} e^{-2\pi i nk/M}.
\label{s14}
\eeq
Next define another unitary operator, $V$, that shifts the eigenvectors of
$U$ cyclically, but in the opposite direction
\beq
V \vert u_n \rangle = \vert u_{n-1} \rangle, \qquad n \not=1,
\qquad
V \vert u_1 \rangle = \vert u_{M} \rangle . 
\label{s15}
\eeq
The same methods, with $U$ replaced by $V$, give
\beq
V^M=I
\label{s16}
\eeq
\beq
V \vert v_m \rangle = v_m \vert v_m \rangle
\qquad v_m = e^{2 \pi i m \over M}
\label{s17}
\eeq
\beq
\vert v_n \rangle \langle v_n \vert
=
{1 \over M} \sum_{m=0}^{M-1} ({V \over v_n})^m =
{1 \over M} \sum_{m=1}^{M} ({V \over v_n})^m
\label{s18}
\eeq
and for unit normalized $\vert v_n \rangle$
\beq
\vert \langle u_k \vert v_n \rangle\vert = {1 \over \sqrt{M}}.
\label{s19}
\eeq
The phase of the $\vert v_n \rangle$ is defined by choosing
\beq
\langle u_M \vert v_n \rangle = {1 \over\sqrt{M}} .
\label{s20}
\eeq
With this choice of phase
\beq
\langle u_M \vert v_n \rangle \langle v_n \vert u_k \rangle =
\langle v_n \vert u_k \rangle {1 \over \sqrt{M}} =
{1 \over M} \sum_{m=0}^{M-1} v_n^{-m} \langle u_m\vert u_k \rangle =
{1 \over M} v_n^{-k} 
\label{s21}
\eeq
which gives
\beq
\boxed{
\langle v_k \vert u_n \rangle = {1 \over \sqrt{M}} e^{-2\pi i nk/M}.
}
\label{s22}
\eeq
Comparing (\ref{s14}) and (\ref{s22}) it follows that
\beq
\vert v_k \rangle =\sum_{m=0}^{M-1} \vert u_m \rangle \langle u_m \vert v_k \rangle=
\sum_{m=0}^{M-1} \vert u_m \rangle {e^{2 \pi i nk/M}\over \sqrt{M}} = 
\sum_{m=0}^{M-1} \vert u_m \rangle \langle u_m \vert a_k \rangle =
\vert a_k \rangle
\label{s23}
\eeq
so the operators $A$ and $V$ have the same eigenvectors.
The unitary operators $U$ and $V$ defined above satisfy
\[
UV = U \sum_{m=0}^{M=1-1} \vert v_m \rangle e^{i2 \pi m \over M}
\langle v_m \vert =
\sum_{m=0}^{M-1} \vert v_{m+1} \rangle e^{i2 \pi m \over M}
\langle v_m \vert =
\]
\beq
\sum_{m=0}^{M-1} \vert v_{m+1} \rangle e^{i2 \pi m \over M}
\langle v_{m+1} \vert U= 
\sum_{m=0}^{M-1} \vert v_{m+1} \rangle e^{i2 \pi (m+1) \over M}
\langle v_{m+1} \vert U=
e^{-2 \pi i \over M} VU
\label{s24}
\eeq
or
\beq
\boxed{
UV = VU e^{-2\pi i \over M}.
}
\label{s25}
\eeq
$U$ and $V$ form an irreducible set of operators in the sense that 
that any
operator on the Hilbert space can be expressed as a degree $(M-1) \times (M-1)$ polynomial
these two operators.  To show this note
\[
\vert v_m \rangle \langle v_k \vert   =
U^{m-k} \vert v_k \rangle \langle v_k \vert   =
\]
\beq
{1 \over M} \sum_{n=0}^{M-1} e^{-2 \pi i nk  /M} U^{m-k} V^n =
{1 \over M} \sum_{n=0}^{M-1} e^{-2 \pi i mn /M} V^n U^{m-k}
\label{s26}
\eeq
where (\ref{s25}) was used to change the order of the $U$ and $V$ operators
in (\ref{s26}).   Irreducibility follows since
a general operator $O$ can be expressed in terms of its matrix elements
in a basis
\[
O = \sum_{m,k=0}^{M-1} \vert v_m \rangle \langle v_m \vert O \vert v_k \rangle
\langle v_k \vert =
\sum_{m,k=0}^{M-1} \langle v_m \vert O \vert v_k \rangle
\vert v_m \rangle  \langle v_k \vert =
\]
\beq
{1 \over M} \sum_{m,n,k=0}^{M-1} e^{-2 \pi i nk /M} \langle v_m \vert O \vert v_k \rangle
U^{m-k} V^n =
{1 \over M} \sum_{m,n,k=0}^{M-1} e^{-2 \pi i mn /M}\langle v_m \vert O \vert v_k \rangle V^n U^{m-k}.
\label{s27}
\eeq
These equations have the form
\beq
\boxed{
O = \sum_{m,n=0}^{M-1} a_{mn} U^m V^n = \sum_{m,n=0}^{M-1} b_{mn} V^m U^n 
}
\label{s28}
\eeq
which is the Weyl representation of $O$. 
If $O$ commutes with $U$ then
\beq
0=  \sum_{mn=0}^{M-1} a_{mn} [U^m V^n,U] = \sum_{mnk=0}^{M-1} a_{mn} U^{m+1} V^n
(e^{2\pi i n/M}-1)
\label{s29}
\eeq
which requires $n=M$ or $0$. This means $O$ is independent of $V$.  Similarly if
$O$ commutes with $V$ it must be independent of $U$.  This
means that any operator that commutes
with both $U$ and $V$ is a constant multiple of the identity.

The following property will be used in the discussion of complex
probabilities
\beq
\boxed{
\sum_{m=0}^{M-1} \langle u_n \vert v_m\rangle =
{1 \over \sqrt{M}} \sum_{m=0}^{M-1} e^{2 \pi i mn\over M} = \delta_{n0}\sqrt{M} =
\delta_{nM} \sqrt{M}
}.
\label{s30}
\eeq
To prove this consider two cases.
If $n=0$ or $M$ the sum is $M$.  Otherwise
\beq
\sum_{m=0}^{M-1} e^{{2\pi i mn \over M}} =
{1 - e^{2 \pi i n}
\over 1 - e^{2 \pi i n/M}} =0  \qquad 0<n<M .
\label{s31}
\eeq

\section{qbits}\label{qbit}

One property of the Schwinger representation is that
it has a natural representation in terms of q-bits.  When $M$ can be
factored into products of prime numbers the $U$ and $V$ can be replaced
by an algebra of commuting pairs of operators with cycles the length
of each prime factor.  The case of most interest for quantum computing
is when $M=2^L$.  In that case the irreducible algebra is represented
by a product of q-bit gates.

To show this assume that $M=2^L$ for large $L$.  The indices $n= 0 \cdots 2^L-1$
can be labeled by $L$ numbers that can only take the values $0$ and $1$:
$n \leftrightarrow (n_1,n_2,\cdots, n_L)$
\beq
n=\sum_{m=1}^L n_m 2^{m-1}. 
\label{qb1}
\eeq
This results in the identifications
\beq
\vert u_{n_1\cdots n_L} \rangle :=\vert u_n \rangle
\qquad
\vert v_{n_1\cdots n_L} \rangle :=\vert v_n \rangle .
\label{qb2}
\eeq
Define unitary operators $U_i$ and $V_i$ by
\beq
U_i \vert v_{n_1\cdots n_L} \rangle =
\vert v_{n_1\cdots [n_i+1]_{\mbox{mod\,2}} \cdots n_L} \rangle
\label{qb3}
\eeq
\beq
V_i \vert u_{n_1\cdots n_L} \rangle =
\vert u_{n_1\cdots [n_i-1]_{\mbox{mod\,2}} \cdots n_L} \rangle .
\label{qb4}
\eeq
Applying what was done in the general case to $M=2^L$ gives
\beq
U_i^2-1= V_i^2 -1=0,
\label{qb5}
\eeq
\beq
[U_i,U_j]=[V_i,V_j]=0 \qquad [U_i,V_j]=0 \qquad i \not=j \qquad
V_i U_i = U_i V_i e^{i \pi}
\label{qb6}
\eeq
\beq
U^n = \prod_{m=1}^L U_m^{n_m}
\label{qb7}
\eeq
\beq
V^n = \prod_{m=1}^L V_m^{n_m}.
\label{qb8}
\eeq
Since $U$ and $V$ can be constructed from the $U_i$ and $V_i $ the
set of $\{ U_i\}$ and $\{ V_i\}$ is also irreducible.

A simple matrix representation of $U_i$ and $V_i$ is
\beq
V_i = \sigma_3 \qquad U_i = \sigma_1
\label{qb9}
\eeq
which are simple quantum gates.
In this representation, $v_0=u_0=1; v_1=u_1=-1$ and
\beq
\vert v_0 \rangle =
\left(
\begin{array}{c}
1 \\
0\\
\end{array}
\right )
\qquad
\vert v_1 \rangle =
\left (
\begin{array}{c}
0 \\
1\\
\end{array}
\right )
\label{qb10}
\eeq

\beq
\vert u_0 \rangle ={1 \over \sqrt{2}}
\left(
\begin{array}{c}
1 \\
1\\
\end{array}
\right )
\qquad
\vert u_1 \rangle = {1 \over \sqrt{2}}
\left (
\begin{array}{c}
1 \\
-1\\
\end{array}
\right ).
\label{qb10}
\eeq
The operators  $\sigma_1$ and $\sigma_3$ 
\beq
U_i=
\sigma_1 = \left (
\begin{array}{cc}
  0&1\\
  1&0\\
\end{array}
\right )
\qquad
V_i= \sigma_3 = \left (
\begin{array}{cc}
  1&0\\
  0&-1\\
\end{array}
\right )
\qquad
\label{qb11}
\eeq
satisfy (\ref{s2}) and (\ref{s15}) for $M=2$.
They also satisfy
\beq
\sigma_3 \sigma_1 = \sigma_1 \sigma_3 e^{{2\pi i \over 2}}
\qquad
(\sigma_1^2 -1)= (\sigma_3^2-1) =0 .
\label{qb12}
\eeq
Any linear operator $A$ on this 2-dimensional
vector space is a polynomial with constant coefficients $a_i$ in these
operators:
\beq
A = a_1I + a_2 \sigma_1 + a_3 \sigma_3 +a_4 \sigma_3\sigma_1 .
\label{qb13}
\eeq
This shows how the discrete version of the irreducible
Weyl algebra can be built up out
of q-bits using the two elementary gates $\sigma_1$ and $\sigma_3$
acting on each qbit.  This means that any operator on the $2^L$
dimensional Hilbert space can be expressed as a
polynomial in the $L$ pairs of 2 state $U$ and $V$ operators.

Note for $M$ odd the same construction works with
$3\times 3$ matrices with
\beq
U_i = \left (
\begin{array}{ccc}
  0&0&1\\
  1&0&0\\
  0&1&0
\end{array}
\right )
\qquad
V_i = \left (
\begin{array}{ccc}
1&0&0\\
0&e^{2 \pi i/3}&  0\\
0 & 0 & e^{4 \pi i /3}
\end{array}
\right ).
\label{qb14}
\eeq

\section{Schwinger's continuum limit}\label{cont}

The eigenvalue spectrum of many observables of interest, like momenta and
coordinates, are continuous.
It is possible to use the discrete
algebra generated by $U$ and $V$ to make a discrete approximation to
the continuum in the large $M$ limit.  To do this assume $M$ is large
and define the small quantity $\epsilon$ by
\beq
%\boxed{
\epsilon^2 :=
{2 \pi /M}.
%}
\label{cl1}
\eeq
For the purpose of approximating the continuum it is convenient
(but not necessary) to
choose $M=2K+1$ odd and number the eigenvectors and eigenvalues from
$-K \leq n \leq K$ instead of $0$ to $M-1$ or $1$ to $M$.
Discrete approximations to continuous
variables $p$ and $q$ are defined by
\beq
p_l = l \epsilon = l \sqrt{2 \pi \over M}   \qquad q_l = l \epsilon = 
l \sqrt{2 \pi \over M} \qquad  -K\epsilon \leq q_l,p_l \leq K\epsilon
\label{cl2}
\eeq
where
\beq
K\epsilon  = \sqrt{M\pi \over 2} - \sqrt{\pi \over 2M}.
\label{cl3}
\eeq
With these definitions the separation between successive values of $p_l$
and $q_l$, $p_{l+1}-p_l =q_{l+1}-q_l = \epsilon$
vanishes as $M \to \infty$ while at the
same time the maximum and minimum values of $p_l$ and $q_l$,
$p_{\pm K} = q_{\pm K} = \pm (
\sqrt{M\pi \over 2} - \sqrt{\pi \over 2M} )$ approach $\pm\infty$ in
same limit.

While for finite $M$ any vector with finite elements has a finite norm - in the
continuum limit ($M\to \infty$) this is no longer true so
the limiting vectors with finite norm should be square summable.
This means that components of vectors with large $\vert l \vert$ should
approach $0$ in the $M\to \infty$ limit.

For $U$ and $V$ given by (\ref{s2}) and (\ref{s15})
Hermitian operators $\hat{p}$ and $\hat{q}$ are defined by
\beq
\boxed{
V = e^{i \epsilon \hat{p}}
\qquad
U= e^{i \epsilon \hat{q}}
} .
\label{cl4}
\eeq
These can be used to define
\beq
V(q_m) = e^{i\hat{p} q_m} = e^{i\hat{p} \epsilon m} = V^m
\label{cl5}
\eeq
\beq
U(p_n) = e^{i\hat{q} p_n} = e^{i\hat{q} \epsilon n} = U^n .
\label{cl6}
\eeq
With these definitions equation (\ref{s25}) becomes
\beq
V(q_m)U(p_k)= U(p_k)V(q_m) e^{i2 \pi mk \over M}=
U(p_k)V(q_m) e^{i \epsilon m \epsilon k}=
U(p_k)V(q_m)e^{ip_kq_m}
\label{cl7}
\eeq
\beq
\boxed{
V(q_m)U(p_k)= 
U(p_k)V(q_m)e^{ip_kq_m}
}
\label{cl8}
\eeq
which is the Weyl \cite{Weyl} form of the canonical commutation
relations, where in this
case the variables are discrete.
Equations (\ref{cl5}-\ref{cl6}) motivate the definitions
\beq
%\boxed{
dp = \epsilon dn = \sqrt{2 \pi \over M} dn
\qquad
dq = \epsilon dn = \sqrt{2 \pi \over M} dn .
%}
\label{cl9}
\eeq
It follows from (\ref{cl4}) that eigenvectors of $V$ are also
eigenvectors of $\hat{p}$ and the eigenvectors of $U$ are also
eigenvectors of $\hat{q}$.  Choosing normalization of the state $\vert
p_n\rangle$ and $\vert q_n \rangle$ so
\beq
\int dp \approx \sum_{l=-K}^K {dp \over dl} = \epsilon \sum_{l=-K}^K
\qquad
\int dq \approx \sum_{l=-K}^K {dq \over dl} = \epsilon \sum_{l=-K}^K
\label{cl9}
\eeq
\beq
I = \sum_{l=-K}^K  \vert v_l \rangle \langle v_l \vert =
\sum_{l=-K}^K  \vert p_l \rangle dp_l \langle p_l \vert =
\sum_{l=-K}^K  \vert p_l \rangle \epsilon \langle p_l \vert
\label{cl10}
\eeq
\beq
I = \sum_{l=-K}^K  \vert u_l \rangle \langle u_l \vert =
\sum_{l=-K}^K  \vert q_l \rangle dq_l \langle q_l \vert =
\sum_{l=-K}^K  \vert q_l \rangle \epsilon \langle q_l \vert 
\label{cl11}
\eeq
and comparing these equations leads to the definitions
\beq
%\boxed{
\vert p_l \rangle := \vert v_l \rangle /\sqrt{\epsilon}
%}
\label{cl12}
\eeq
and 
\beq
%\boxed{
\vert q_l \rangle := \vert u_l \rangle /\sqrt{\epsilon} .
%}.
\label{cl13}
\eeq
Using these relations gives
\beq
\langle p_m\vert q_n \rangle = 
        {1 \over \epsilon} \langle v_m \vert u_n \rangle =
        {1 \over \epsilon \sqrt{M}}e^{-2 \pi i mn\over M}
        =
        {1 \over \sqrt{2 \pi}} e^{-i p_m q_n}
\eeq
\beq
\langle p_m\vert p_n \rangle = 
{1 \over \epsilon} \langle v_m \vert v_n \rangle =
{\delta_{mn} \over \epsilon}
\label{cl14}
\eeq
and
\beq
\langle q_m\vert q_n \rangle = 
{1 \over \epsilon} \langle u_m \vert u_n \rangle =
{\delta_{mn} \over \epsilon}.
\label{cl15}
\eeq

A result that will be used later to reinterpret the path
integral as the expectation of a potential functional with respect to
a complex probability distribution follows from (\ref{s30}). Consider
the expression
\beq
\sum_{lm} \langle q_n \vert p_l \rangle dp_l f(p_l) \langle p_l \vert q_m
\rangle dq_m =
{\epsilon^2 \over 2\pi }\sum_{l=-K}^K\sum_{m=-K}^K
e^{i (q_n -q_m) p_l } f(p_l) .
\label{cl16}
\eeq
The $m$ sum can be computed in closed form
\beq
\sum_{m=-K}^K e^{i q_m p_l} = \sum_{m=-K}^K e^{2 \pi ml \over M} =
e^{-i \pi l K \over M}\left ({1 \over 1- e^{2 \pi l \over M}}
- {e^{2 \pi l}\over 1- e^{2 \pi l \over M}}\right ) =
\delta_{l0}M .
\label{cl17}
\eeq
Using (\ref{cl17}) in (\ref{cl16}) with (\ref{cl1}) gives
\beq
\boxed{
\sum_{lm} \langle q_n \vert p_l \rangle dp_l f(p_l) \langle p_l \vert q_m
\rangle dq_m =
{M \epsilon^2 \over 2\pi  }f(0) = f(0).
}
\label{cl18}
\eeq
The same result is obtained by ``integrating'' over the final $q_n$ instead of
the initial $q_m$.  This result will be used in the development of
discrete path integrals that follows.

\section{complex probabilities}\label{complex}

A complex probability system is defined
by a sample set $S$ and a complex valued function $P$ on subsets of
$S$ with the
properties
\beq
P(S_i) = \sum_{s\in S_i} P(s) \qquad 
P(S) =1 .
\label{cp1}
\eeq
$P(S_i)$ is the complex probability assigned
to the subset $S_i$ of $S$.
It follows that 
\beq
P(S_i)+P(S_i^c)= 1
\label{cp2}
\eeq
where $S_i^c$ is the complement of $S_i$ in $S$, 
and for a finite set of non-intersecting subsets of $S$
\beq
S_i\cap S_j = \mbox{$\empty$ } i \not=j  \qquad P(\cup S_i) =\sum_i P(S_i).
\label{cp3}
\eeq
In the applications that follow the sample set will be a finite collection
of paths.

More generally, since $P(s)$ is complex, equation (\ref{cp3}) cannot
be extended to countable non-intersecting subsets, which is where
complex probabilities differ from ordinary probabilities
\cite{Muldowney2}\cite{Muldowney}.  This is not an issue for finite
sample sets.  The extension of the notion of complex probabilities to
continuous a sample set generated from intervals by complements and
finite unions, based on the Henstock integral
\cite{Henstock}\cite{bartle}\cite{Gill}, was used in \cite{Katya_1}\cite{Katya_2}
to prove that the real-time path integral formulated as the expectation of a
potential functional with respect to a complex probability
distribution on cylinder sets of paths converges to a global solution of the Schr\"odinger
equation.  This was applied to compute scattering amplitudes using
real-time path integrals in a simple model in \cite{polyzou}.  In that
case the complex probability was a probability on a finite collection
of cylinder sets rather than a discrete set of paths.  To treat the
large number of cylinder sets, the probability was approximately
factored into products of one-step probabilities, which reduced the
problem to computing powers of approximate transfer matrices.  Because
of the approximation unitarity was only preserved approximately.  In
the discrete case the sample set is finite, the complex probability
exactly factors into a product of one time step complex probabilities
and the transfer matrices associated with the one-step probabilities
are exactly unitary.

A random variable $F(s)$ is a function on the sample set
$S$ with expectation value
\[
E[F] = \sum_{s \in S}  P(s) F(s) .
\]
In this paper the sample set is the finite collection of paths that
have $M$ possible values at each of $N$ time steps, the complex
probability, $P(s)$ is associated with free propagation through $N$
time steps along the path ``$s$'', and $F(s)$ is the contribution from
the potential due to the path ``$s$''.  This is discussed in the next section.

\section{Complex probabilities in real time path integrals}\label{path}

The path integral for a system with one degree of freedom is
formulated using the discrete representation discussed section \ref{weyl}.
Following references \cite{Muldowney}\cite{Katya_1}\cite{Katya_2},
the path integral will be defined as the expectation value of a
potential functional with respect to a complex probability
distribution.

To do this it is necessary to:
\begin{itemize}
\item[1.)] Define the space of paths
\item[2.)] Define complex probabilities on the space of paths
\item[3.)] Identify the path integral with the expectation value of a
functional on the space of paths.
\end{itemize}

Let $H$ be a canonical Hamiltonian with one degree of freedom of the form
\beq
H= {\hat{{p}}^2 \over 2\mu} + V(\hat{q})
\label{pi1}
\eeq
where $\hat{q}$ and $\hat{p}$ are canonically conjugate operators satisfying
\beq
[\hat{q},\hat{p}] =i .
\label{pi2}
\eeq
The starting point for constructing a path integral is
the Trotter product formula \cite{Simon}\cite{Katya_3}:
\[
\langle q_f, t_f \vert e^{-i H t } \vert q_i ,t_i \rangle =
\langle q_f, t_f \vert (e^{-i H t/N })^N \vert q_i ,t_i \rangle =
\]
\beq
\lim_{N \to \infty} \langle q_f, t_f \vert (e^{-i H t/N })^N \vert q_i ,t_i \rangle
=
\lim_{N \to \infty}  \langle q_f, t_f \vert (e^{-i (\hat{p}^2 / 2\mu)\Delta t }
e^{-i V(\hat{q})\Delta t}
)^N \vert q_i ,t_i \rangle
\label{pi3}
\eeq
where $\Delta t := t/N$.  This is the operator generalization of
the representation
\beq
e^x = \lim_{N\to \infty} (1+{x\over N})^N 
\label{pi3a}
\eeq
of $e^x$.  Equation (\ref{pi3}) is exact in the limit that $N\to \infty$
when applied
to a normalizable wave packet.  It is also possible to express $e^{-i
  H \Delta t}$ in terms of the $U$ and $V$ operators
in the Weyl representation (\ref{s28}), however the more familiar
representations are used in this section.

Following the standard steps in evaluating the path integrals, 
sums over complete sets of eigenstates of $U$ and $V$ are inserted
between the operators in (\ref{pi3}):
\[
\langle q_f  \vert e^{-iHt}\vert \psi \rangle = 
\int \langle q_f  \vert e^{-i H t } \vert q_i  \rangle
dq_i \psi(q_i) dq_i =
\]
\[
\lim_{N\to \infty}
\int \langle q_f \vert p_N \rangle dp_N e^{-i (p_N^2/2\mu)\Delta t}
\langle p_{N}\vert q_N \rangle dq_N e^{-i V(q_N) \Delta t} \cdots
\]
\beq
\cdots \langle q_2 \vert p_1 \rangle dp_1 e^{-i (p_1^2 / 2\mu)\Delta t}
\langle p_1 \vert q_1 \rangle dq_1 \langle q_1\vert e^{-i V(q_1)
\Delta t}
\vert \psi \rangle .
\label{pi4}
\eeq
The next step is to approximate the integrals by numerical
quadratures.  This is done using the discrete variables introduced in
the previous section.
%This can also be thought of as modeling a
%system with continuous variables by one with discrete variables.
While this is not the most efficient approximation, it has the
advantage that everything is discrete, finite and exactly unitary.

In this case for an $N$ time step Trotter approximation the discrete
variable $q_i$ for the $i$-th time step can take on the $2K+1$
discrete values $l\epsilon$, $-K \leq l \leq K$ and the transition
amplitude (\ref{pi4}) becomes
\[
\langle q_f  \vert e^{-iH(t)}\vert \psi \rangle \approx
\sum \langle q_f \vert p_{Nn_N} \rangle \epsilon e^{-i (p_{Nn_N}^2/2\mu)\Delta t}
\langle p_{Nn_N}\vert q_{Nn_N} \rangle \epsilon e^{-i V(q_{Nn_N}) \Delta t} \times
\]
\[
\langle q_{Nn_N} \vert p_{N-1n_{N-1}} \rangle \epsilon e^{-i (p_{N-1n_{N-1}}^2/2\mu)\Delta t}
\cdots
\langle p_{Nn_N}\vert q_{Nn_N} \rangle \epsilon e^{-i V(q_{Nn_N}) \Delta t} \times
\cdots
\]
\beq
\langle q_{2n_2} \vert p_{1n_1} \rangle \epsilon e^{-i (p_{1n_1}^2 / 2\mu) \Delta t} \langle
p_{1n_1}
\vert q_{1n_1} \rangle \epsilon e^{-i V(q_{1n_1})\Delta t}
\langle q_{1n_1} \vert \psi_i
 \rangle .
\label{pi5}
\eeq
The next step is to ``integrate'' over the ``momentum'' variables.
While this can be done
exactly for quadratic functions of $p$ in terms of Fresnel integrals,
here this integral is 
replaced by a finite sum.  The first step is to define a one time step
free propagation operator.  This is the $q$-space representation of the
transfer matrix for free propagation \cite{Feynman_1}\cite{Feynman_2}
for a time $\Delta t$:
\[
K(q'_{m},  q_{n},\Delta t ) dq_{n}:=
\sum_{l=-K}^K 
\langle q_{m} \vert p_{l} \rangle
\epsilon e^{-i (p_{l}^2 / 2\mu) \Delta t} \langle
p_l \vert q_{n} \rangle \epsilon =
\]
\beq
{\epsilon^2 \over 2 \pi} \sum_{l=-K}^K
e^{i(q_{m}'- q_{n})p_l -i (p_{l}^2 / 2\mu) \Delta t}.       
\label{pi6}
\eeq
It follows from (\ref{cl18}) that the integral over the initial
coordinate is $1$:
\beq 
\int K(q',q,\Delta t) dq \to
\sum_{n=-K}^K  K(q_m,q_n,\Delta t)\epsilon   = 1
\label{pi7}
\eeq
independent of $q_m$.  Because of this, $K(q_m,q_n,\Delta t) dq$, is
interpreted as the complex probability for making a transition from
state $q_n$ to state $q_{m}$ in time step $\Delta t$.  The probability
interpretation follows because the sum is $1$. In this case the sample set of
probabilities is finite.  The interpretation of equation (\ref{pi7}) is
that a state
that ends up at $q_m$ has to have started at one of the $2K+1$ $q_n$'s
with complex probability 1.

The path integral (\ref{pi5}) can be expressed in terms of (\ref{pi7}) as
\[
\langle q_f \vert e^{-i H t } \vert q_i , \rangle \approx
\]
\[
\sum_{n_1 \cdots n_N} K(q_f,q_{N,n_N}, \Delta t) e^{-iV(q_{Nn_N})\Delta t} \epsilon
K(q_{N,n_N},q_{N-1,n_{N-1}}, \Delta t) e^{-iV(q_{N-1,n_{N-1}})\Delta t} \epsilon
\cdots
\]
\[
\cdots K(q_{2,n_2},q_{1,n_{1}}, \Delta t) e^{-iV(q_{1,n_1})\Delta t} \epsilon=
\]
\[
\sum_{n_1 \cdots n_N} K(q_f,q_{N,n_N}, \Delta t) \epsilon
K(q_{N,n_N},q_{N-1,n_{N-1}}, \Delta t) \epsilon
\cdots
K(q_{2,n_2},q_{1,n_{1}}, \Delta t) \epsilon \times
\]
\beq
e^{-i\sum_{l=1}^N V(q_{ln_l})\Delta t}.
\label{pi8}
\eeq
This is expressed as finite powers of products of finite-dimensional unitary
transfer matrices.

Define 
\[
P_N(q_f,q_{N},q_{N-1}, \cdots ,q_{2}, q_{1} ):=
\]
\beq
K(q_f,q_{N}, \Delta t) \epsilon K(q_{N},q_{N-1} \Delta t) \epsilon
\cdots
K(q_{3},q_{2}, \Delta t) \epsilon 
K(q_{2},q_{1}, \Delta t) \epsilon
\label{pi9}
\eeq
which represents free propagation from $q_1$ to $q_f$ along a path
through
$q_2, q_3, \cdots q_N$.
By (\ref{pi7}) it follows that summing over all of $q_{i,n_{i}}$ gives 1
independent of $q_f$,
\beq
\sum_{n_1,\cdots, n_N}P_N(q_f,q_{N},q_{N-1}, \cdots ,q_{2}, q_{1} ):=1
\eeq

It is now possible to define the space of paths between $q_i$ and
$q_f$.  A path $\gamma$ is a $N$-dimensional vector $(q_1,q_2,\cdots
q_N)$ where each of the $q_n$ can have one of the $M$ discrete eigenvalues
of $q$.  This vector represents a
path that starts at $q_1=n_1\epsilon$ and after time $\Delta t$ is at $q_2=n_2\epsilon$,
$\cdots$ , and after $N-1$ time steps is at $q_N=n_N\epsilon$ and arrives at $q_f$
after $N$ time steps.  The set of all $M^N$ paths that end up at
$q_f$ is denoted by $\Gamma$.

The quantity $P_N(\gamma) := P_N(q_f; q_N ,q_{N-1}, \cdots ,q_{2},
q_{1} )$ is a complex number that is interpreted as the complex
probability for a particle to travel on the path $\gamma =q_1 \to q_2
\to \cdots \to q_{N-1} \to q_N \to q_f$ since
\beq
\sum_{\gamma \in \Gamma} P_N(\gamma)=1 .
\label{pi10}
\eeq
It is the discrete analog of the complex probability that path lies
in a given cylinder set of paths.

For the path $\gamma$ a potential ``functional''
of the path $\gamma$ is defined by
\beq
W[\gamma] =e^{-i\sum_{n=1}^N  V(q_{n})\Delta t}
\label{pi11}
\eeq
where the sum is over each of the $q_n\in \gamma$.  With 
this notation the approximate transition amplitude is
\[
\langle q_f, t_f \vert e^{-i H t } \vert q_i ,t_i \rangle \approx
\]
\[
\sum_{n_1,n_2, \cdots, n_{N}=1}^M
P_N(q_f,q_{Nn_N},q_{N-1 n_{N_1}}, \cdots ,q_{2 n_2}, q_{1n_1} )
\times
e^{-i\sum_{k=1}^N V(\epsilon n_k)\Delta t} \delta_{q_{1n_1},q_i} =
\]
\beq
\sum_{\gamma\in \Gamma}P_N(\gamma) W[\gamma]\delta_{q_{1n_1},q_i}
\label{pi12}
\eeq
which is represented by the expectation $E[W\delta]$ of the potential functional
$W[\gamma]\delta_{q_{1n_1},q_i}$ with respect to the
complex probability distribution $P_N(\gamma)$.

Note that this transition amplitude can be expressed exactly as
the $N$-th power 
\beq
X^N_{f j} \delta_{ji}
\label{pi13}
\eeq
of the transfer matrix
\beq
X_{ij} := K_{ij}W_j 
\label{pi14}
\eeq
where
\beq
K_{ij} := 
K(q_j,q_k, \Delta t)\epsilon
\qquad
W_j := e^{-iV(q_j) \Delta t} 
\label{pi15}
\eeq
applied to the initial state.  The important observation is that
even though there are $(2K+1)^N$ discrete paths, the discrete path
integral involves computing the $N^{th}$ power of a single $(2K+1)\times
(2K+1)$ dimensional matrix.

%While the computation is reduced to matrix algebra, it can be deconstructed to
%find the complex probability of each of the $M^N$ paths in $\Gamma$.

It is interesting to note that while the computation of the path integral
is reduced to matrix multiplication, the matrix product can be deconstructed to find the contribution of
each path to $E[W\delta]$.

In \cite{polyzou} a similar method was used to compute sharp momentum
scattering transition matrix elements
using real-time path integrals interpreted as expectation values of
a potential functional with respect to a complex probability
distribution on cylinder sets of paths.  In that application the factorization of the complex
probability into a product of one time step
probabilities was only approximate and as a result unitarity was
only satisfied approximately.

%The method was applied to scattering by a short-range
%local Gaussian potential in one dimension.

In \cite{polyzou} sharp momentum scattering matrix elements
were approximated using a path integral approximation to
matrix elements of the scattering 
transition operator
%\beq
%\langle \Psi_F \vert  V
%X^{N}(K^{\dagger})^N] \vert \Psi_I \rangle .
%\eeq
%which approximates
\beq
T_{fi}=   \lim_{t \to -\infty} \langle k_f \vert V e^{-Ht}e^{-iH_0t}
\vert k_i\rangle. 
\label{pi16}
\eeq
The sharp momentum eigenstates $\vert k_{f/i} \rangle$ were replaced by
normalizable states $\vert \Psi_I \rangle$ and $\vert \Psi_F \rangle$
that are sharply peaked about the initial and final momenta respectively and
have spatial support in the interaction volume.  They were normalized
like delta functions in the sense that they integrate to 1.

%In the discrete case the energy is not exactly conserved, so
%this matrix element has to be computed in normalizable states
%with the same approximate energy.
In the form (\ref{pi16}) the interaction $V$ provides a convenient volume cutoff
on the localized initial scattering state.  The matrix
$(K^{\dagger})^N X^{2N}(K^{\dagger})^N$, which converges to the
scattering operator, is exactly unitary so it can
be diagonalized with eigenvalues of the form $e^{2 i \delta_n}$,
where $\delta_n$ are eigenvalues of a phase shift operator.

The convergence of this method depends on the convergence of the
Trotter limit and the convergence of the discrete quadrature.
Mathematically the Trotter product formula converges strongly
for suitable Hamiltonians; this was used in \cite{Katya_2} to show that in
the continuum case the expectation of the potential functional with
respect to the complex probability associated with free propagation
converges to global solutions of the Schr\"odinger equation.
This suggests that final result is independent of the order of making the
Trotter approximation and the discrete quadrature approximation.

This interpretation of the path integral as the expectation value of a
random variable over a complex probability on a space of paths has a
conceptual advantage over the conventional interpretation.  In the
conventional interpretation of the path integral in terms of the
action functional, the finite difference representation of the
``derivatives'' in the action involves differences that never get
small as the time steps get small, rendering the interpretation of the
path integral as an integral over paths weighted by a ``measure''
depending on the action questionable.  The representation in terms of
complex probabilities involves a real potential functional defined on
continuous paths.  The potential functional can be thought of as a
{\it perturbation of a complex Gaussian process} associated with
free propagation.

\section{scattering in the discrete representation}\label{scatt}

Formal scattering theory is an idealization.  A real scattering experiment
takes place in a finite volume during a finite time interval.  The relevant
physics is dominated by a finite number of degrees of freedom that are
limited by the energy and scattering volume.

The fundamental quantum mechanical observable is the probability
for a transition from a prepared initial state to a detected final state
\beq
P_{fi} = \vert \langle \psi_f (t)\vert \psi_i (t) \rangle \vert^2.
\label{sca1}
\eeq
While the individual states depend on time, the probability
(\ref{sca1}) is independent of $t$ due to the unitarity of the time
evolution operator.  The important consideration is that both states have to be
evaluated at the same time.  The problem of scattering theory is that
there is no common time when both the initial and final states are
simple.  On the other hand the initial state is simple before the
collision and the final state is simple after the collision.

The initial and final states at the time of collision can be
determined by evolving them from times where they behave like
non-interacting subsystems to the collision time.  Since localized
wave packets spread, the effects of spreading can be eliminated by
starting with localized wave packets at the collision time,
evolving them beyond the range of
interactions using free time evolution, and then evolving them back to the
interaction region using the full Hamiltonian.  The result is a
unitary mapping that transforms the free wave packet at the collision
time to the dynamical wave packet at the same time.

If $U_0(t)$ and $U(t)$ represent the free and dynamical unitary
time evolution operators,
then assuming the time of collision is approximately at time $t=0$ the scattering
asymptotic conditions have the form
\beq
\Vert \vert U(\pm \tau)\vert \psi_{\pm}(0) \rangle -
U_0(\pm \tau) \vert \vert \psi_{0\pm}(0) \rangle \Vert 
\approx 0
\label{sca2}
\eeq
where $\tau$ is sufficiently large for the interacting particles to be
separated beyond the range of their mutual interactions.  This
expression is independent of $\tau$ for sufficiently large $\tau$, but
the minimum value of $\tau$ depends on the range of the interaction
and the structure of $\vert \psi_{0\pm}(0) \rangle$.  Normally dependence
on these conditions is removed by taking the limit $\tau \to
\infty$.  In this work, for computational reasons, it is desirable to
choose $\tau$ as small as possible, which requires paying attention to
the range of the interaction and the structure of the initial and
final states.

The unitarity of the time evolution operator means that (\ref{sca2})
can be replaced by
\beq
\Vert  \vert \psi_{\pm} (0)\rangle -
U(\mp \tau)U_0(\pm \tau) \vert \psi_{0\pm}(0) \rangle \Vert 
\approx 0 .
\label{sca3}
\eeq
The operators
\beq
\Omega_{\pm}(\tau) := U(\pm \tau)U_0(\mp \tau)
\label{sca4}
\eeq  
are unitary mappings from $\vert \psi_{0\pm} (0)\rangle$ to
$\vert \psi_{\pm}(0) \rangle$.

Using these definitions the scattering probability can be expressed as
\beq
P_{fi} = \vert \langle \psi_{0+}(0)\vert  S(\tau) \vert \psi_{0-}(0) \rangle \vert^2 \label{sca5}
\eeq
where
\beq
S(\tau) := \Omega^{\dagger}(\tau)\Omega (-\tau)
\label{sca6}
\eeq
is the scattering operator.
Since $S(\tau)$ is unitary it can be expressed in terms of a self-adjoint
phase shift operator
\beq
S(\tau) = e^{2i \delta (\tau)}
\label{sca6}
\eeq
where $S(\tau)$ should be independent of $\tau$ for sufficiently
large $\tau$.

In a real experimental measurement the probability (\ref{sca5})
depends on the structure of the initial and final wave packets, which
cannot be precisely controlled by experiment.  If the matrix elements
of $S(\tau)$ in sharp momentum states are slowly varying functions of
momentum, then the dependence on the wave packet factors out
\cite{brenig} and can be eliminated to compute differential cross
sections.  In this case the sharp momentum matrix elements can be
approximated from the matrix elements using Gaussian (minimal
uncertainty) wave packets with a ``delta-function normalization'' that
are sharply peaked about the desired momenta.

%In the large $\tau$ limit sharp momentum matrix elements of $S(\tau)$
%exactly conserves energy; in a calculation using finite $\tau$ energy
%conservation is only approximate - so states should be chosen so they
%have the same mean energy.

This formulation of scattering is amenable to a path integral
treatment.  As previously discussed scattering reactions are dominated
by a finite number of degrees of freedom.  The use of the discrete
Weyl representation has the advantage that unitarity is exactly
preserved on truncation to a finite number of degrees of freedom.
Alternative path integral treatments of scattering appear in
\cite{Campbell}\cite{Rosenfelder1}\cite{Rosenfelder2}.

The advantage of the discrete representation is that 
$U_0(-\tau)U(2\tau)U^0(-\tau)$ can be expressed as the limit of
products of the transfer matrices defined in the previous section
\beq
S(\tau) = \lim_{N\to \infty} K^{-N}X^{2N} K^{-N}
\label{sca7}
\eeq
where
\beq
K_{ij}= K(q_i,q_j,\Delta t)\epsilon
\qquad
(X)_{ij}= K(q_i,q_j,\Delta t)\epsilon e^{-i V(q_j)\Delta t},
\label{sca8}
\eeq
$\Delta t = \tau/N$ and $N$ is the number of Trotter time slices. 
Note also that
\beq
K^{N} = K(f_f,q_i,N\Delta t).
\eeq

Sharp-momentum matrix elements of the scattering operator can be expressed in
terms of the matrix elements of the transition operator $T$, which is
easier to calculate in the discrete representation
\beq
S= I - 2 \pi i \delta (E_f -E_i) T
\label{sca9}
\eeq
where $T$ is approximately given by  
\beq
T_s \approx V \Omega (-\tau)
\label{sca10}
\eeq
when evaluated in normalizable states with sharply peaked momenta.
The advantage of this representation is that for scattering problems
$V$ is a short range operator that provides a volume cutoff.

In the discrete representation sharp momentum eigenstates are
normalizable however {\it they cannot be used in scattering calculations}
because they are completely delocalized in space because the
discrete momenta and coordinates are complementary
- making it impossible to get to the asymptotic region.

The most straightforward way to construct suitable initial or final wave packets
in the discrete representation is to approximate the corresponding
minimal uncertainty states of the continuum theory.  The quantities
to control are the mean position,  momentum and the uncertainty
in both of these quantities defined for a given state $\vert \psi \rangle$
by:
\beq
\langle q \rangle_{\psi} := \sum_{n=-K}^K {\langle \psi \vert u_n \rangle
  n \epsilon \langle u_n \vert \psi \rangle \over
  \langle \psi \vert \psi \rangle }
\qquad
\langle p \rangle_{\psi} := \sum_{n=-K}^K {\langle \psi \vert v_n \rangle
n \epsilon \langle v_n \vert \psi \rangle \over
  \langle \psi \vert \psi \rangle }
\label{sca12}
\eeq
\beq
(\Delta q)^2 =\langle \psi \vert (q-\langle q \rangle)^2 \vert \psi \rangle =
\sum_{n=-K}^K {\langle \psi \vert u_n \rangle
((n \epsilon)^2 - \langle q \rangle^2 )\langle u_n \vert \psi \rangle \over
  \langle \psi \vert \psi \rangle }
\label{sca13}
\eeq
 \beq
(\Delta p)^2 = \langle \psi \vert (p-\langle p \rangle)^2 \vert \psi \rangle=
\sum_{n=-K}^K {\langle \psi \vert v_n \rangle
((n \epsilon)^2 - \langle p \rangle^2 )\langle v_n \vert \psi \rangle \over
\langle \psi \vert \psi \rangle }.
\label{sca14}
\eeq
The continuum delta function normalized minimal uncertainty states are
\beq
\langle p \vert \psi_{0}  (0) \rangle =
{1 \over 2 \sqrt{\pi} \Delta p} e^{- {(p-\langle p \rangle )^2 \over 4 (\Delta p)^2}} .
\label{sca15}
\eeq
where $\langle p \rangle$ is the mean momentum and
$\Delta p$ is the quantum mechanical uncertainty in
$p$ for this wave packet.  This wave packet needs to be evolved to
$-\tau$ using the free time evolution which adds a phase to (\ref{sca15}): 
\beq
\langle p \vert \psi_{0} (-\tau) \rangle =
{1 \over 2 \sqrt{\pi} \Delta p} e^{ - {(p-p_i)^2 \over 4 (\Delta p)^2}
  + i {p^2 \over 2 \mu} \tau}.
\label{sac16}
\eeq
In the discrete ``$p$'' representation this is replaced by 
\beq
\langle n \vert \psi_{0}(-\tau) \rangle = C e^{-{(\epsilon n - \langle p \rangle)^2 \over
4 (\Delta p)^2} + i {n^2 \epsilon^2 \over 2 \mu} \tau} .
\label{sac17}
\eeq
where $C$ is a normalization constant.
In the $x$ representation this becomes
\beq
\langle m_q \vert \psi_0 \rangle = {\epsilon \over \sqrt{2\pi}}
\sum_{n=-K}^K e^{i \epsilon^2 mn}
\langle n \vert \psi_{0}(-\tau) \rangle .
\label{sac18}
\eeq

To illustrate that this gives a good approximation to the continuum
results  $\langle p \rangle$,
$\langle p \rangle$, $\Delta q$ and $\Delta p$ were calculated starting
with $\langle p \rangle=2.5$, $\Delta p=.25$ and $K=300$ as input parameters in
(\ref{sac16}).  The results of the calculation
\begin{align}
\mbox{mean}_{p-calc}&=2.500000\\
\mbox{mean}_{q-calc}&=-2.51 \times 10^{-17}\\
\Delta_{p-calc}&=.3000000\\
\Delta_{q-calc}&= 1.666667
\end{align}
are consistent with the input parameters,  the minimal uncertainty condition
$\Delta p \Delta q = 1/2$,
and the continuum results.

As a test
the discrete approximation
was applied to the problem of one-dimensional scattering of
particle of mass $m$ by a repulsive Gaussian potential of the form
\beq
V(q) = \lambda e^{-\alpha q^2}
\eeq
with $\lambda=.5$ and $\alpha =2.0$.
The potential is plotted in figure 1.  The particle's mass is taken to
be 1 in dimensionless units so the velocity and momentum can be
identified.  The initial wave packet is a Gaussian with a delta
function normalization in momentum space with mean momentum $p=2.5$
and width $\Delta p= .25 $. It is pictured in figure 2.  The Fourier
transform of the initial wave packet is given in figure 3.  The oscillations
are due the to fact that the momentum space wave packet has a non-zero mean momentum.  Given the
size of the potential and wave packets, the wave packet needs to move
about 18 units to the left in order to be out of the range of the
potential.  This suggest that for $v=p/m=2.5$ that $\tau=7$ should
be sufficient to move the wave packet out of the range of the
potential.  The resulting
free wave packet at $\tau=-7$ is shown in figure 4.  The scattered
wave function with $K=300$ ($M=601$) after $N=100$ time steps is shown
in figure 5,
and that
result multiplied by the potential is shown in figure 6.
Compared to the wave function in figure 3, the wave function in figure 5
includes the effects of the interaction.
Figure 6 illustrates the cutoff due to the short range potential; it
illustrates how only the part of the wave function inside the range of the
interaction contributes to the scattering operator.
Figure 7 compares the result of the off-shell Born approximation $\langle p
\vert V \vert \psi(0) \rangle$ to the calculation of the real and
imaginary parts of $\langle p \vert T \vert \psi(0) \rangle$ while
figure 8 compares $\langle p \vert T \vert \psi(0) \rangle$ to
$\langle p \vert T (p_0)\vert p_0 \rangle$ obtained by numerically
solving the Lippmann-Schwinger equation using the method \cite{Rubtsova}.

Figure 8 shows that the path integral computation with an initial wave
packet with a width of 1/10 of the momentum converges to the numerical
solution of the integral equation.  In unrelated time-dependent
scattering calculations \cite{Kopp:2011vv} a $\Delta p$ of about a tenth of
$p$ gave good approximations to sharp momentum matrix elements of the
transition operator for a wide range of momenta.

Unlike the solution of
the Lippmann Schwinger equation, in the path integral approach for
each energy it is necessary to determine minimal values of
$M$,$N$,$\tau$ and $\Delta p$ that are needed for convergence.
In practice there are a number of trade offs.  Making
the wave packets narrow in momentum increases the scattering volume
in the coordinate representation.  This in turn requires a larger $\tau$
to get out of the range of the potential.  If $\tau$ gets too large
the wave packet can move past $q_{max}=K\epsilon$ and will reappear
at $q_{min}=-K\epsilon$.  As $p$ gets large the oscillations in the
$q$ space wave function have higher frequencies, which requires
smaller time steps, while when $p$ gets small it is necessary to
make the wave packet width in momentum small enough so the
coordinate space tail of the wave function
gets out of the interaction volume.

The computations require storing the initial vector.  It is not
necessary to store the transfer matrix - it can be computed
efficiently on the fly.  This is important for realistic calculations
since the vectors will be significantly larger in higher dimensions.
The hope is that in the future q-bits can be used to represent large
vectors.

This one-dimensional example approximated half-shell
sharp-momentum transition matrix elements.  The on-shell values can be
used to extract other observables such as phase shifts and in the
one-dimensional case transmission and reflection coefficients.  This
formulation of the one-dimensional problem in terms of transition
operators has the advantage that the method can be formally extended
to treat a large class of scattering problems.

The formulation of the discrete path integral used a reducible
discrete Schwinger representation where the complex one time step
probability is represented by a dense matrix.  An equivalent
irreducible representation in terms of qbits involves a product of
matrices (\ref{qb7}-\ref{qb8}) that act on single qbits, which may
have computational advantages.

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_1.pdf}
\caption{\bf Potential}
\label{fig:1}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_2.pdf}
\caption{\bf Momentum space initial Gaussian wave packet}
\label{fig:2}
\end{minipage}
\end{figure}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_3.pdf}
\caption{\bf Coordinate space initial Gaussian wave packet}
\label{fig:3}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_4.pdf}
\caption{\bf Free Gaussian wave packet at $\tau=-4$}
\label{fig:4}
\end{minipage}
\end{figure}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_5.pdf}
\caption{\bf Initial scattering state at $t=0$ }
\label{fig:5}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_6.pdf}
\caption{\bf V $\times$ initial scattering state at $t=0$}
\label{fig:6}
\end{minipage}
\end{figure}


\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_7.pdf}
\caption{\bf $\langle p \vert V \vert \psi_{0i}(0) \rangle$}
\label{fig:7}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{fig_8.pdf}
\caption{\bf $\langle p \vert T \vert \psi_{0i}(0) \rangle$}
\label{fig:8}
\end{minipage}
\end{figure}

\section{discrete multi-resolution representation of quantum field theory}\label{qft}

One motivation for studying quantum computing in physics is that it
might provide a framework for a numerical treatment of problems in
quantum field theory.  Clearly this goal is a long way off for
realistic theories, but the state of quantum computing is advancing
rapidly.  Discrete formulations of field theory naturally fit into the
discrete framework discussed in this work and should be relevant
for future applications.

A numerical treatment of quantum field theory requires a truncation to
a system with a finite number of degrees of freedom.  For reactions
that take place in a finite space-time volume and involve a finite
energy it is natural to limit the number of degrees of freedom
by making volume and resolution truncations.  Degrees of freedom that
are outside of this volume or energetically inaccessible are expected
to be unimportant for the given reaction.
Daubechies wavelets \cite{daubechies}\cite{jorgensen1}\cite{jorgensen2} and scaling functions are a
basis for square integrable functions and a natural representation to perform
both kinds of truncations.  The basis consists of a complete
orthonormal set of functions that have compact support and a limited
amount of smoothness.  They have the property that for any small
volume there are an infinite number of basis functions supported
entirely in that volume.  This means that they can be used to
construct ``local'' observables by smearing the fields with basis
functions.  All of the basis functions $\xi_n(x)$ are generated from
the solution of a linear renormalization group equation by
translations and dyadic scale transformations, which facilitates
computations.  Because they are
complete they can be used to {\it exactly} expand canonical fields
\beq
\Phi (\mathbf{x},t) = \sum \Phi_n (t) \xi_n (\mathbf{x}) \qquad
\Pi (\mathbf{x},t) = \sum \Pi_n (t) \xi_n (\mathbf{x})
\eeq
where $\Phi_n (t)$ and $\Pi_n (t)$ are discrete field operators.
If the fields satisfy canonical equal time commutation relations
\beq
[\Phi (\mathbf{x},t), \Pi(\mathbf{y},t)] = i \delta (\mathbf{x}-\mathbf{y})
\eeq
then the discrete fields $\Phi_n$ and $\Pi_n$ will satisfy discrete
versions of the canonical equal time commutation relations
\cite{Bulut:2013bg} \cite{Polyzou:2017wnj}
\cite{Polyzou:2020ifj}: 
\beq
[\Phi_m(t),\Pi_n(t)]=i \delta_{mn} \qquad
[\Phi_m(t),\Phi_n(t)] =0 \qquad
[\Pi_m(t), \Pi_n(t)] =0 .
\label{w1}
\eeq
In terms of these degrees of freedom the Hamiltonian
for a $\phi^4$ theory has the form
\beq
H= {1 \over 2}\sum_n \Pi_n\Pi_n + {m^2 \over 2}\sum_n \Phi_n\Phi_n +
\sum_{mn} D_{mn} \Phi_m \Phi_n + \lambda
\sum_{klmn} \Gamma_{klmn} \Phi_k\Phi_l\Phi_m\Phi_n
\label{wav2}
\eeq
where the sum are all infinite.  Since $H$ commutes with itself the
discrete fields can be evaluated at $t=0$.
The constant matrices are defined
by the integrals
\beq
D_{mn} = 
{1 \over 2} \int \pmb{\nabla} \xi_n (\mathbf{x})\cdot
\pmb{\nabla} \xi_m (\mathbf{x})d\mathbf{x} 
\label{wav3}
\eeq
\beq
\Gamma_{klmn}= \int
\xi_k (\mathbf{x}) \xi_l (\mathbf{x}) \xi_m (\mathbf{x}) \xi_n (\mathbf{x})
d\mathbf{x} .
\label{wav4}
\eeq
For the wavelet basis
these constants vanish unless all of the functions appearing in the
integrals have a common support, which makes them almost local.  In
addition, because all of the functions in the integrand are related by
translations and scale transformations to a single function, the
integrals can all be expressed as linear combinations of solutions of
some small linear systems generated by the renormalization group
equation (\ref{w3}).  Unlike a lattice truncation, the wavelet representation
of the field theory is (formally) exact (before truncation).  The
basis functions regularize the fields so local products of fields that
appear in the Hamiltonian are replaced by infinite sums of
well-defined products of discrete field operators.  The basis functions are
differentiable, so there are no finite difference approximations.

Wavelet representations of quantum field theories have been discussed
by a number of authors
\cite{best:1994}
\cite{federbush:1995}
\cite{1995NuPhB.436..414H}
\cite{Battle:1999}
\cite{best:2000}
\cite{Ismail1:2003}
\cite{Ismail2:2003}
\cite{altaisky:2007}
\cite{albeverio:2009}
\cite{altaisky:2010}
\cite{altaisky:2010}
\cite{altaisky:2013}
\cite{Bulut:2013bg}
\cite{altaisky:2013b}
\cite{PhysRevA.92.032315}
\cite{PhysRevLett.116.140403}
\cite{altaisky:2016b}
\cite{altaisky:2016c}
\cite{altaisky:2016}
\cite{altaisky:2017}
\cite{Polyzou:2017wnj}
\cite{Neuberger2018}
\cite{Tomboulis1}
\cite{Polyzou:2020ifj}
\cite{Altaisky:2021hbq}.

What is relevant is that the Hamiltonian (\ref{wav2}) has the same form
as (\ref{pi1}), except it involves an infinite number of degrees of
freedom.  It is diagonal and quadratic in the discrete momentum
operators and has a non-trivial (almost local) dependence on the
$\Phi_n$ operators.  Because all of the basis functions are constructed
from the fixed point $s(x)$ of the renormalization group equation
(\ref{w3}) the constant quantities
$D_{mn}$ and $\Gamma_{klmn}$ can be expressed in terms of a finite
set of elementary integrals. 

The advantage of this basis is that it has natural volume and
resolution truncations.  For reactions taking place in a finite volume
with a finite energy a (large) finite number of these degrees of
freedom should provide a good approximation.  This reduces the problem
to a problem with a finite number of discrete degrees of freedom.  In
addition the truncated Hamiltonian still has the form (\ref{wav2}),
except the sums are only over the retained discrete modes. As the volume and
resolution are increased (i.e as more modes are added) the parameters of
the theory have to be adjusted to keep the some physical observables
constant.  

The truncated problem is a finite number of degree of freedom
generalization of the one degree of freedom problem discussed in the
section \ref{scatt}.  For a quantum field theory the vector
representing the state of the field will be much larger than in the
one degree of freedom scattering case.

The construction of the wavelet basis used to construct the
discrete representation of the Hamiltonian (\ref{wav2}) is outlined below.
The starting point the solution of the linear renormalization group
equation
\beq
\boxed{
s(x) = \sum_{l=0}^{2L-1}h_l D T^l s(x)
}
\label{w3}
\eeq
%\label{wav.1}
where
\beq
Df(x):= \sqrt{2}f(2x) \qquad \mbox{and} \qquad Tf(x) := f(x-1)
%\label{wav.2}
\label{w4}
\eeq
are unitary discrete dyadic scale transformations and unit
translations.  The $h_l$ are constants that depend on the choice of
$L$. Generally as $L$ increases the solutions, $s(x)$, become smoother
but the support increases.  A useful case is $L=3$ where the
solution $s(x)$ of (\ref{w3}), called the scaling function, has
support on $[0,2L-1]=[0,5]$ and has one continuous derivative.  In
that case the coefficients $h_l$ for the Daubechies $L=3$ scaling
functions are
\[
h_0=(1+\sqrt{10}+\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\[
h_1=(5+\sqrt{10}+3\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\[
h_2=(10-2\sqrt{10}+2\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\[
h_3= (10-2\sqrt{10}-2\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\[
h_4=(5+\sqrt{10}-3\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}
\]
\beq
h_5 =(1+\sqrt{10}-\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2} . 
\label{w5}
\eeq
They are chosen so the solution of (\ref{w3}) and unit translates
are orthonormal and locally finite linear combinations of these
unit translates can
be used to locally pointwise represent degree 2
polynomials.
Given the solution, $s(x)$, of (\ref{w3})
new functions are constructed from $s(x)$
by rescaling and translating
\beq
s^k_n(x) := D^k T^n (x)s(x) = \sqrt{2^k} s(2^k x-n) .
%\label{wav.6}
\label{w6}
\eeq
The starting scale is fixed using
\beq
\int s(x) dx=1 .
\label{w7}
\eeq
The functions $s^k_n(x)$ for fixed $k$ span a subspace of the square
integrable functions on the real line with a resolution $2^{-k}L$:
\beq
{\cal S}^k := \{ f(x) \vert f(x) = \sum_{n=-\infty}^\infty c_n s^k_n(x) \qquad
\sum_{n=-\infty}^\infty  \vert c_n\vert^2 < \infty \}.
%\label{wav.5}
\label{w8}
\eeq
The renormalization group equation (\ref{w3}) implies
\beq
{\cal S}^k \subset {\cal S}^{k+1}.  
%\label{wav.11}
\label{w9}
\eeq
It follows that
\beq
{\cal S}^{k+1} = {\cal S}^{k} \oplus {\cal W}^{k}.    
%\label{wav.12}
\label{w10}
\eeq
where ${\cal W}^k$ is the orthogonal complement of
${\cal S}^k$ in ${\cal S}^{k+1}$.
An orthonormal basis for the subspace 
${\cal W}^k$ is the ``wavelet functions'':
\beq
w^k_n(x)=D^kT^n w(x)
\label{w11}
\eeq
where
\beq
w (x):= \sum_{l=0}^{2L-1} (-)^l h_{2L-1-l} D T^{l} s(x) 
%\label{wav.13}
\label{w12}
\eeq
is called the ``mother wavelet''.
This decomposition can be continued to generate a multi-resolution decomposition of $L^2(\mathbb{R})$
\beq
\boxed{
L^2 (\mathbb{R}) = {\cal S}^{k} \oplus_{l=0}^\infty {\cal W}^{k+l}.
%=
%\oplus_{n=-\infty}^{\infty}   {\cal W}^{n}.
%\label{wav.17}
}
\label{w13}
\eeq
This results in a multi-resolution  orthonormal basis for
$L^2 (\mathbb{R})$
\beq
\{\xi_n(x) \}_{n=-\infty}^{\infty} :=
\{ s^k_n(x)\}_{n=-\infty}^\infty \cup \{ w^m_n(x)\}_{n=-\infty,l=k}^\infty .  
%\label{wav.18}
\label{w14}
\eeq
For the choice $L=3$ the basis functions $s^k_n(x)$ and $w^k_n(x)$ have
compact support on
$[2^{-k}n,2^{-k}(n+5)]$.  All of the basis functions have one
continuous derivative so the coefficients (\ref{wav3}) are defined .  The functions $s^k_n(x)$ are like splines in
that linear combinations can be used to locally pointwise represent
degree 2 polynomials while the functions $w^l_n(x)$ are orthogonal to
the same polynomials on their support.  The Fourier transforms of the
basis functions are entire functions due to their compact support.
Orthonormal three dimensional basis functions are products of
one-dimensional basis functions.   In spite of these nice properties,
the basis functions are fractal valued (since they are related to fixed points
of a renormalization group equation) and cannot be written down in
closed form. 

In order to use this representation the constant coefficients 
$D_{mn}$ and $\Gamma_{n_1 \cdots n_k}$ that appear in the Hamiltonian (\ref{wav2})
need to be computed.
Using scale transformations (\ref{w4}) and the renormalization group
equation (\ref{w3}) they can all be expressed in terms of the integrals
\beq
d_{n} = 
\int {ds(x) \over dx} {ds(x-n) \over dx}dx \qquad -4\leq n \leq 4
\eeq
\beq
\gamma_{m,n,k}= \int s(x) s(x-m) s(x-n) s(x-k) dx \qquad -4 \leq mnk \leq 4 .
\eeq
These integrals are related to each other by finite linear equations
derived from the renormalization
group equation (\ref{w3}) and the scale fixing condition (\ref{w7}).
These linear systems can formally be solved in terms of the coefficients
$h_l$ (\ref{w5}).  The coefficients $d_n$ are rational numbers and
can be found in the literature on wavelets \cite{beylkin1}.
To find the $\gamma_{mnk}$ requires finding eigenvalues of a
$9^3 \times 9^3$ matrix.  This eliminates the need be able to evaluate
fractal valued functions.
Alternatively the integrals $\gamma_{mnk}$ can be approximated by
noting that the renormalization group equation (\ref{w3}) and the
scale fixing condition (\ref{w7})
can be used to exactly
calculate the basis functions and their derivatives exactly at all dyadic
rational points.  Since the functions and their derivatives
are continuous this can be used to estimate these quantities and
integrals involving these quantities to any desired accuracy.

In order to illustrate a path integral treatment of this system
consider a truncation of the theory in 1+1 dimension 
where only 2 adjacent modes of the Hamiltonian (\ref{wav2}) are retained.
In this case the overlap coefficients that appear in the Hamiltonian
and couple adjacent modes can be expressed in terms of the following
quantities
\begin{align}
\Gamma_{0000}&= 0.9528539 \\
\Gamma_{0001}&= 0.0670946 \\
\Gamma_{0011}&= 0.0890895 \\
\Gamma_{0111}&=-0.1424536 \\
D_{00}&=295./56.; \\
D_{01}&=-356./105.; \\
D_{10}&=D_{01};\\
D_{11}&=D_{00}   
\end{align}
where the $\Gamma$ coefficients were computed by numerical integration
using the trapezoidal rule with the basis functions evaluated
at 256 dyadic points on their support.  Convergence was verified 
using 512 dyadic points. 

The truncated Hamiltonian in this case is
\beq
H= {1 \over 2}\sum_{n=0}^1 \Pi_n\Pi_n + {m^2 \over 2}\sum_{n=0}^1 \Phi_n\Phi_n
+\sum_{mn} D_{mn} \Phi_m \Phi_n + \lambda
\sum_{klmn=0}^1 \Gamma_{klmn} \Phi_k\Phi_l\Phi_m\Phi_n
\label{wavxx}
\eeq
where
$\Gamma_{0000}=\Gamma_{1111}$,  $\Gamma_{0001}=\Gamma_{0010}=\Gamma_{0100}=
\Gamma_{1000}$, etc. .
The path integral treatment of the field theory in the discrete representation
is a multi-dimensional generalization of the treatment for one degree of freedom
where each field mode represents an independent degree of freedom.

A general numerical treatment involves a truncation and
renormalization
followed by two approximations.  The truncation discards all but a
finite number, $F$, of discrete degrees of freedom.
\beq
H \to H_F
\label{w22}
\eeq
Ideally physics at a given energy scale and in a given volume should
be dominated by a finite number of accessible degrees of freedom.  The
remaining degrees of freedom that are not expected to impact
calculation at that given scale and volume are discarded.  The
truncated theory is renormalized by adjusting the parameters of the
theory so a set of observables agree with experiment.  This gives the
parameters a dependence on the choice of retained degrees of freedom.
This is a truncation rather than an approximation. It assumes that no
additional parameters need to be introduced beyond what appears in the
truncated Hamiltonian and that there is a limit as the volume becomes
infinite and resolution becomes arbitrarily small.  This is followed
by two approximations.  The first approximation is to approximate the
unitary time evolution operator for the truncated theory using the
Trotter product formula with $N$ time slices.
\beq
U_F(\tau) = e^{-i H_F \tau} =
\lim_{N\to \infty} (e^{-i H_F(\Pi)\Delta t}e^{-iH_F(\Phi)\Delta t})^N 
\label{w23}
\eeq 
where $\Delta t = \tau/N$ and
\beq
H_F = H_F(\Pi) + H_F(\Phi)
\label{w24}
\eeq
with
\beq
H_F(\Pi) := {1 \over 2}\sum_n \Pi_n\Pi_n
\label{w24a}
\eeq
and
\beq2
H_F(\Phi):=
 {m^2 \over 2}\sum_n \Phi_n\Phi_n +
\sum_{mn} D_{mn} \Phi_m \Phi_n + \lambda
\sum_{klmn} \Gamma_{klmn} \Phi_k\Phi_l\Phi_m\Phi_n
\label{w24b}
\eeq
which expresses $H_F$ as the sum of a part with only the $\Pi_n$
fields and another part with only the $\Phi_n$ fields.  Since the
discrete canonical pairs of field operators $\Phi_n$ and $\Pi_n$
satisfy canonical commutation relations they have a continuous
spectrum on the real line.  This is because each one of these
complementary operators generates translations in the other operator.
The last step is to approximate the
continuous spectrum of the discrete field operators $\Phi_n$ and
$\Pi_n$ by a collection of $M=2K+1$ closely spaced eigenvalues
$\phi_n,\pi_n = n \epsilon$ where $-K \leq n \leq K$ and $\epsilon^2 =
2 \pi/M$.  This is exactly what was done in the one-dimensional case,
except in this case there are $F$ degrees of freedom where $F$ is the
number of retained discrete field modes.  Unlike the truncation,
both of these steps are mathematical approximations.  

Let $\langle \pmb{\phi} \vert \chi \rangle =
\chi(n_1\epsilon, \cdots , n_F \epsilon) $ be a localized function of the
amplitudes of the $F$ discrete field modes that represent an initial free wave
packet.

The goal is to use path integrals to calculate the time evolution of
these coupled modes.  For the field theory, before truncation, in the
discrete representation there are integrals over an infinite number of
modes.  For the discarded modes the volume being integrated over for
each mode is infinite, resulting in an infinite product of an infinite
number of infinite irrelevant constants.
The advantage of discretizing the integrals is
that the volume for each mode is finite:
\[
\mbox{Volume}= \sqrt{2 M\pi}- \sqrt{2 \pi /M}.
\]
The discarded modes can be
eliminated by summing and dividing by this finite volume, mode by mode,
before taking the
continuum limit.  In this
way the integrals over discarded degrees of freedom are replaced by a product of 1's.
This results in a path integral that only involves
the retained degrees of freedom.  The discrete approximation results
in a sample space with a finite number of discrete paths.

The Trotter approximation is 
\[
\langle
n_1,n_2,\cdots n_F \vert U_F(\tau) \vert \chi (0) \rangle =
\]
\beq
\lim_{N\to \infty} \langle
n_1,n_2,\cdots n_F \vert  (e^{-i H_f(\Pi)\Delta t}e^{-iH_F(\Phi)\Delta t})^N
\vert \chi (0) \rangle .
\label{w25}
\eeq
This can be evaluated by inserting complete sets of eigenstates of
the complementary fields 
between each of the operators.  The following abbreviations
are used for sums over intermediate states:
\beq
\int d\pmb{\phi} = \epsilon^F \sum_{n_1=-K}^K \cdots  \sum_{n_F=-K}^K,
\label{w26}
\eeq
for vectors representing a value of the eigenvalues of each of the $F$
independent $\phi$
fields, 
\beq
\pmb{\phi}=(n_1 \epsilon, \cdots , n_F \epsilon) \qquad -K \leq n_i \leq K, 
\label{w27}
\eeq
for vectors representing the value of the eigenvalues of each of the $F$
independent $\pi$
fields
\beq
\pmb{\pi}=(n_1 \epsilon, \cdots , n_F \epsilon) \qquad -K \leq n_i \leq K 
\label{w28}
\eeq
and
\beq
\gamma = (\pmb{\phi}_0,\pmb{\phi}_1, \cdots, \pmb{\phi}_N)
\label{w29}
\eeq
for a ``path'' that ends at $\pmb{\phi}_0$ where
$\pmb{\phi}_j$ ($j>0$) 
represents values of each of the $\phi_n$ fields at
each of $N$ time steps.

The following definitions are generalizations of the definitions in
section \ref{path}:
\beq
K(\pmb{\phi}',\pmb{\phi},\Delta t) := \sum_{\mathbf{n}''}
\langle \pmb{\phi}' \vert \pmb{\pi} \rangle \Delta t
\langle \pmb{\pi} \vert \pmb{\phi} \rangle .
\label{w30}
\eeq
It follows from (\ref{cl18}) that $K(\pmb{\phi}',\pmb{\phi},\Delta t)$ has the property
\beq
\sum_{\mathbf{n}} K(\pmb{\phi}',\pmb{\phi},\Delta t)\epsilon^F =1
\label{w31}
\eeq
and
\[
P(\pmb{\phi}_f,\pmb{\phi}_N, \cdots \pmb{\phi}_1) :=
\]
\beq
K(\pmb{\phi}_f,\pmb{\phi}_N,\Delta t)\epsilon^F
K(\pmb{\phi}_N,\pmb{\phi}_{N-1},\Delta t)\epsilon^F
\cdots 
K(\pmb{\phi}_3,\pmb{\phi}_2,\Delta t)\epsilon^F
K(\pmb{\phi}_2,\pmb{\phi}_1,\Delta t)\epsilon^F
\label{w32}
\eeq
also satisfies
\beq
\sum_{\gamma \in \Gamma}  
P(\pmb{\phi}_f,\pmb{\phi}_N, \cdots , \pmb{\phi}_1) =1 .
\label{w33}
\eeq
Equation (\ref{w32}) represents the complex probability of a given path, where at
each time slice each of the $F$ $\phi$'s has one of $M$ allowed values
between $-K\epsilon$  and $K\epsilon$.  Removing the last factor
of $\epsilon^F$ and only summing over $\pmb{\phi}_N \cdots \pmb{\phi}_2$
gives the evolution due to free propagation
\beq
\langle
\pmb{\phi}_f \vert e^{-{i\over 2} \pmb{\Pi}\cdot \pmb{\Pi}\tau}
\vert \pmb{\phi}_1 \rangle =
\sum_{\pmb{n}_n \cdots \pmb{n}_1} 
P(\pmb{\phi}_f,\pmb{\phi}_N, \cdots , \pmb{\phi}_1)\epsilon^{-F} .
\label{w34}
\eeq
The full path integral including the effects of the interaction can
be expressed as the expectation of the following potential functional
of the path $\gamma$ with respect to the complex probability
distribution (\ref{pi12}):
\beq
W[\gamma] :=
e^{i\sum_{n} H_F (\pmb{\phi}_n)   \Delta t}
\label{w36}
\eeq
where
$H_F (\pmb{\phi}_n)$ represents the value of the $\phi$-dependent part
of the Hamiltonian evaluated at the value of the path $\gamma$ at the
$n$-th time slice.

This gives the path integral approximation
\[
\langle
n_{1f},n_{2f},\cdots n_{Ff} \vert U_F(\tau) \vert \chi (0) \rangle =
\]
\beq
\sum_{\gamma} 
P(\pmb{\phi}_f,\pmb{\phi}_N, \cdots \pmb{\phi}_1)W[\gamma]
\chi (\pmb{\phi}_1)
\label{w37}
\eeq
which again represents the path integral for fields as the expectation value
of
a potential functional with respect to a
complex probability distribution.  As in the one degree of freedom case
this can be exactly factored into a product of one-time step operators
\[
P(\pmb{\phi}_f,\pmb{\phi}_N, \cdots \pmb{\phi}_1)W[\gamma] =
\]
\[
K(\pmb{\phi}_f,\pmb{\phi}_N,\Delta t)
e^{iH_F (\pmb{\phi}_N)   \Delta t}\epsilon^F
K(\pmb{\phi}_N,\pmb{\phi}_{N-1},\Delta t)
e^{iH_F (\pmb{\phi}_{N-1})   \Delta t}\epsilon^F
\cdots
\]
\beq
K(\pmb{\phi}_3,\pmb{\phi}_2,\Delta t)
e^{iH_F (\pmb{\phi}_2)   \Delta t}  \epsilon^F
K(\pmb{\phi}2,\pmb{\phi}_1,\Delta t)
e^{iH_F (\pmb{\phi}_1)   \Delta t}\epsilon^F .
\label{w38}
\eeq
This can be used to represent time evolution as the product of
large approximate transfer matrices.

At each stage these calculations use finite mathematics.   The use
of the finite Weyl representation exactly preserves unitary at each
level of approximation.  Both the $\pmb{\phi}$ and $\pmb{\pi}$
transfer matrices are unitary and can be expressed exactly in the
truncated model.  This means that the discrete Trotter approximation to
time evolution is exactly unitary.

The calculation shown in figures nine and ten show the initial
real and
imaginary parts of the two field modes.  In this case the
initial modes are real and taken to be Gaussians of the form
\beq
\langle \phi_1,\phi_2 \vert
\psi \rangle =
N e^{-\sum_{i=0}^1 (\phi_i - \langle \phi_i \rangle)^2/(4 \delta \phi_i^2)}
\eeq 
Figures 11 and  12 show the real and imaginary parts of 
the time $t=.5$ evolved amplitudes of these two
discrete modes with $M=41$ values using 
$N=20$ Trotter steps.

Figures 13 and 14 show plots of the real and imaginary parts of $\phi_0$ when
$\phi_1=0$ at $T=0$ and $T=.5$.  

In the initial calculations the initial mean displacement and
uncertainty of each mode was taken to be .5. The initial state has no
imaginary part but one develops due to the non-zero displacement of
the initial state.  This truncation is too crude to contain any real
physics, however it illustrates the application of the discrete path integral
to fields.

A more drastic truncation of the discretization of the continuum could
be used to explore the dynamics of fields with a larger number of modes.

% calculations performed using daub.c
% graphs daub_r.gp, daub_i.gp unicode greek letters
%
\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.6]{fig_10.pdf}
\caption{\bf Two modes (real) at T=0}
\label{fig:9}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.6]{fig_9.pdf}
\caption{\bf Two modes (imaginary) at T=0}
\label{fig:10}
\end{minipage}
\end{figure}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.6]{fig_11.pdf}
\caption{\bf Two modes (real) after T=.5}
\label{fig:11}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.6]{fig_12.pdf}
\caption{\bf Two modes (imaginary) after T=.5}
\label{fig:12}
\end{minipage}
\end{figure}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.6]{fig_13.pdf}
\caption{\bf One mode (real) after T=0.,.5}
\label{fig:13}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.6]{fig_14.pdf}
\caption{\bf One mode (imaginary) after T=0.,.5}
\label{fig:14}
\end{minipage}
\end{figure}
 
\section{summary and conclusion}\label{sum}

This paper discusses a path integral treatment of discrete
representations of quantum theory.  The treatment is motivated by a
textbook treatment \cite{schwinger} of measurement theory of quantum
systems on finite dimensional Hilbert spaces.  The discrete
representation provides a natural connection to a q-bit representation
in terms of an irreducible set of quantum gates. It was also shown to
formally provide a discrete path integral treatment of problems in
potential scattering and quantum field theory.  The discrete
Weyl representation is closely related to the quantum Fourier
transform while the equivalent decomposition into irreducible sub
algebras is more directly related to quantum circuits.

The treatment starts by considering a general quantum observable with
a finite number of outcomes.  It is used to construct a pair of
unitary operators, one that commutes with the original observable and
a second complementary unitary operator.  The two unitary operators
are a finite dimensional version of the irreducible Weyl algebra on
the Hilbert space spanned by the eigenvectors of the original
operator. When the dimension of the Hilbert space gets large this
algebra approximates the Weyl algebra of a continuum theory.  When the
large number is a power of 2 the algebra can be decomposed into a
product of irreducible sub-algebras where the complementary unitary
operators are elementary qbit gates, which are the building blocks of
quantum circuits.

In the limit of large dimensions discrete operators that behave like
canonical coordinates and momenta can be constructed from this
algebra.  In this approximation the ``coordinates'' and ``momenta''
take on a finite number of discrete values that get closer together
and cover more of the real line as the number of degrees of freedom
increases.

Hamiltonians that are sums of an operator that is quadratic in the
``momentum'' variables and an operator that is a multiplication
operator in the ``coordinate'' variables are considered.  Time
evolution is represented by a product of transfer matrices for a large
number of small steps.  For small time steps the transfer matrix can
be approximately factored into a product of a transfer matrix
involving the ``momentum'' part of the Hamiltonian and another
transfer matrix involving the ``coordinate'' part of the Hamiltonian.
Both of these transfer matrices are represented in the discrete
``coordinate'' representation.  A path is defined to go through one of
the discrete coordinates at each time step.  In the discrete
representation the number of possible paths is $M^N$ where $N$ is the
number of time steps and $M$ is the number of discrete coordinates at
each time slice.

The transfer matrices involving the momentum part of the Hamiltonian
have the property that summing over either the initial or final
coordinates gives 1, independent of the other coordinate.  In this work
the momentum transfer matrix is interpreted as the complex probability
for a transformation from one of the allowed coordinates to another one in
time step $\Delta t$.  The product of $N$ of these operators, where
the final coordinate of one is the initial coordinate of the next one is
interpreted as a complex probability for a given ``path'' on the finite
sample set
of discrete paths.  This probability has the property that summing
over all paths with a given starting point or a given end point is 1.
The interaction (coordinate dependence) is included by multiplying
this probability by the product of the coordinate transfer matrices
evaluated at each point on the path.  In this interpretation the
coordinate contribution is represented by a functional on the space of
paths.  Taking the expectation value of this functional with respect
to the complex probability distribution gives the usual Trotter
product representation of finite time evolution of the discrete system.

In the discrete representation all of the operators are exactly
unitary and the mathematics is finite.  The sample space of paths for
the complex probabilities is finite.  The general structure of the
Hamiltonian as the sum of a quadratic form in the momentum variables
and an interaction term is realized in non-relativistic quantum
mechanics and relativistic quantum field theory.  The application to
potential scattering was discussed using the example of a particle
scattering from a smooth short range interaction in one dimension.  In
the case of field theories an exact multi-resolution representation of
the field in terms of an infinite number of discrete modes was used.
When truncated to a finite number of modes the resulting discrete
system has the structure of system of coupled particles.

The long term interest is in quantum computing.  The examples were
computed by applying products of the one step transfer matrices to an
initial vector.  By computing the transfer matrix elements on the fly,
is was not necessary to store the transfer matrix.  However as the
number of degrees of freedom is increased, the size of the vector
representing the state of the quantum system is the major limitation.

The author would like to thank William Hester for pointing out some errors
in the original version of this manuscript.
\bibliography{quant_comp.bib}
\end{document}


\cite{Gill}
\cite{Rubtsova}
Rosenfelder1,
Rosenfelder2
Campbell
Halliday:1994fe,
Best:1999mu,

Kopp:2011vv,
Federbush:1995rt,


\section{eigenvalue problem}

Of interest in field theory are vacuum correlation functions.
In the discrete representation
\[
\langle 0 \vert \phi(x) \phi(y) \vert 0 \rangle = 
\]
\[
\sum_{\mathbf{m}\mathbf{n}} \xi_{\mathbf{n}}(\mathbf{x})
\xi_{\mathbf{m}}(\mathbf{y}) \langle 0 \vert \phi_{\mathbf{n}}(0) \phi_{\mathbf{m}}
(0,y^0-x^0) \vert 0 \rangle = 
\]
\beq
\sum_{\mathbf{m}\mathbf{n}} \xi_{\mathbf{n}}(\mathbf{x})
\xi_{\mathbf{m}}(\mathbf{y}) \langle 0 \vert
\phi_{\mathbf{n}}(0) U^{y^0-x^0} \phi_{\mathbf{m}}
(0) \vert 0 \rangle 
\label{ev1}
\eeq  
The one quantity that cannot be computed using path integral methods
is the vacuum vector.

In the truncated theory where $F$ is finite there is no translationally
invariant vacuum vector.  The expectation is that for sufficiently large
$F$ that the lowest energy eigenstate of $H$ should provide a
good local approximation to the vacuum that can be used to approximate
correlation functions provided the correlations do not exceed the
volume or resolution of the truncated theory.

Using the notation of the last section wave functions in the discrete
representation have the form:
\beq
\langle \pmb{\phi} \vert \psi \rangle = {1 \over (2 \pi)^{N/2}}
\sum_{\mathbf{n}}  f(\mathbf{n})  e^{-i \epsilon \mathbf{n} \cdot \pmb{\phi}}.
\label{ev2}
\eeq
The eigenvalue problem for a Hamiltonian of the form
\beq
H = {1 \over 2} \pmb{\Pi}^2 + V(\pmb{\phi})
\label{ev3}
\eeq
is given by the linear system for the coefficient $f(\mathbf{m})$:
\beq
\sum_{\mathbf{m}} e^{-i \epsilon \mathbf{m}\cdot \pmb{\phi}}(E- \epsilon^2 \mathbf{m}^2 )f(\mathbf{m})
 =
\sum_{\mathbf{k}\mathbf{n}} 
V(\epsilon\mathbf{n})
e^{i \epsilon^2(\mathbf{k}-\mathbf{m})\cdot \mathbf{n}}
f(\mathbf{k}).
\label{ev4}
\eeq
While this is a large linear system.  The most efficient approach it to
solve it using variational methods.

\beq
\pmb{\phi} = (\phi_1,\phi_2, \cdots , \phi_N)
\eeq
\beq
\sum_{\mathbf{n}} := \sum_{n_1=-K}^K \cdots \sum_{n_N=-K}^K
\eeq
In the discrete approximation $\mathbf{\phi}$ can take on the values
\beq
\pmb{\phi} = (n_1 \epsilon,n_2 \epsilon,\cdots , n_N \epsilon) = \mathbf{n}\epsilin
\eeq
where $-K\leq n_i \leq K$ and the volume element is
\beq
d\pmb{\phi}:= \epsilon^N
\eeq

Wave functions in the $\phi$ representation have the form 


Given the solution for the ground state
correlation functions are defined by
\beq
<0 \vert \phi(x) \phi(y) \vert 0 \vert >=\sum \xi_{\mathbf{l}(x) \xi_{\mathbf{ }(y)
f(\mathbf{m}^* ????
f(\mathbf{n})e^{i \epsilon^2 \mathbf{k}\cdot (\mathbf{m}-\mathbf{n})}

in this representation



\section{correlation functions}

In the field theoretic case the canonical commutation relations provide
an irreducible algebra of operators.  In order to perform computations
it is necessary to also have a representation of the Hilbert space.
For a single degree of freedom this can be done by construction
creation and annihilation operators out of the canonical operators and
defining a ''vacuum vector'' as being annihilated by the annihilation
operator.  Then general vectors can be computed by applying elements of
the itteducible algebra to the ''vacuum''.
\[
a_{\alpha} = {1 \over \sqrt{2}}(\alpha \phi_n +{i \over \alpha} \pi_n)
a^{\dagger}_{\alpha} = {1 \over \sqrt{2}}(\alpha \phi_n -{i \over \alpha} \pi_n)
\]
where $\alpha$ is any non=zero constant.  These operators satisfy
\[
[a_{\alpha},a_{\alpha}^{\dagger}]=1
\]
independent of $\alpha$.  These relations can be inverted to express
$\phi_n$ and $\pi_n
\[
\phi_n = {1\over \alpha \sqrt{2}}(a_\alpha + a_\alpha^{\dagger}) 
\]
\[
\pi_n = {i\over \alpha \sqrt{2}}(a_\alpha^{\dagger} -a_\alpha) 
\]
The creation and annihilation operators corresponding to different values of
$\alpha$ are related by canonical transformations
\[
a_{\alpha'}={1 \over 2} ({\alpha' \over \alpha}+\alpha \over \alpha'})a_{\alpha}
+{1 \over 2} ({\alpha' \over \alpha}-\alpha \over \alpha'})a^{\dagger}_{\alpha}
\]
\[
a^{\dagger}_{\alpha'}=
{1 \over 2} ({\alpha' \over \alpha}-\alpha \over \alpha'})a_{\alpha} +
{1 \over 2} ({\alpha' \over \alpha}+\alpha \over \alpha'})
a^{\dagger}_{\alpha}
\]
These relations have the from
\[
a_{\alpha'}=\cosh (\eta) a_{\alpha} + \sinh(\eta) a^{\dagger}_{\alpha}
\]
\[
a^{\dagger}_{\alpha'}=
\sinh(\eta)a_{\alpha} + \cosh(\eta) a^{\dagger}_{\alpha}
\]
These operator are related by a unitary transformation
\[
a_{\alpha'} = Wa_{\alpha} W^{\dagger} \qquad 
W= e^{i\eta G}  \qqaud G = i {\eta \over 2} (a_{\alpha}^{\dagger} a_{\alpha}^{\dagger}
-
a_{\alpha}a_{\alpha})
\]
Since these are unitarily related in the finite number of degree of
freedom case, any of these vacuua can be used to generate a
representation of the Hilbert space.  In general non of these vacuua
correspond to the ground state of the truncated Hamiltonain, but they
can be used to construct a basis where the Hamiltonian can be
diagonoalized.

If we write the un normalized ground state in the form

\vert g \rangle = e^{-S} \vert 0 \rangle

where $S= \sum_n S_n$ and $S_n$ is an $n$-th degree polynomial in the
$a_{\alpha}^{\dagger}$ the eigenvalue problem can be expressed as
\[
e^{S} (H-E)e^{-S} \vert 0 \rangle =0 =
\]
\[
(H-E + [S,H] + {1 \over 2!} [S,[S,H]] +
  {1 \over 3!} [S,[S,[S,H]]] + {1 \over 4!} [S,[S,[S,[S,H]]]]+ \cdots) \vert 0 \rangle
\]
If $H$ is a polynomial of degree 4 then $S$  
If we take multiply by $\langle 0 \vert,\cdots ,
\langle 0 \vert$,\cdots a_{n_1}a_{n_2} \cdots a_{n_k}$.  When this is done
the only non-zero terms involve at most 4 factors of $S$ with a
total of $n\leq 4$ creation operators>  The equations have the form

They are non-linear
Vacuum expectation values only picks up constant terms

[H_2 S_n ] = (n-1)(H-1)

<i+j| H_i S_j |0>

H0 S1 S2 S3 S4

H1 S1 S1

<1|0> pic up terms


S_1 one creation op a 

S_1 S_1 2 creation op  aa

S1 S1 S1  3 creation operators aaa

S1 S1 S1 S1 4 creation operators aaaa

a S_2 2 annihilation operators a^\dagger a

a S_2   a^\dagger a^{\dagger}

aa S_2 S_1 a^{\dagger}

aa


aaa

aaaa

The truncated field theoretic Hamiltonain involves a finite number of
canonical variables.  In addition the kinetic term is quadratic in the
$\Pi_n$  fields.  In this sense it looks line a large number of copies
of the one degree of freedom case, and can be formally treated using the
same methods.  One of the desirable quantities to compute are
Wightman functions.  In addition to the fields these require a
representation of the vacuum. This is the lowest energy state of the
Hamiltonian.  THe 

In the continuuum case given the Weyl relations the projection on the vacuum
state
\[
\Pi{\alpha}: = {1 \over 2 \pi}\int e^{- {\alpha p^2 \over 4} - {q^2 \over 4 \alpha}}
U(p) V(q) e^{{i \over 2}pq} dp dq
\]
In the discrete case this becomes

\Pi_{\alpha} = {1 \over 2 \pi}\int e^{- {\alpha n^2\epsilon^2 \over 4} - {m^2 \epsilon^2 \over 4 \alpha}}
U(n\epsilon) V(m\epsilon) e^{{i \over 2}mn\epsilon^2} epsilon^2 =

{1 \over 2 \pi}\int e^{- {\alpha n^2\epsilon^2 \over 4} - {m^2 \epsilon^2 \over 4 \alpha}}
U^n V^m e^{{i \over 2}mn\epsilon^2} epsilon^2 =

{1 \over M} \sum_{m,n=-K}^K
U^n V^m
e^{- {\alpha n^2\epsilon^2 \over 4} - {m^2 \epsilon^2 \over 4 \alpha} +{i \over 2}mn\epsilon^2}

If we square this 

\section{commutator}

The action of $V$ and $U$ on eigenstates is given by
\beq
V\vert v_n\rangle = e^{2 \pi i n \over M}\vert v_n\rangle
= e^{i \epsilon^2 n} \vert v_n\rangle = e^{i \epsilon p } \vert
v_n\rangle
\eeq
\beq
U\vert u_n\rangle = e^{2 \pi i n \over M}\vert
vu_n\rangle = e^{i \epsilon^2 n} \vert u_n\rangle = e^{i \epsilon q }
\vert u_n\rangle .
\eeq
These relations define the operators $p$ and
$q$ up to a phase.  The phase can be fixed by defining the operators
by their action on basis functions
\beq p \vert v_n \rangle = n
\epsilon \vert v_n \rangle \qquad q \vert u_n \rangle = n \epsilon
\vert u_n \rangle ,
\eeq
which is consistent with (\ref{}).
\[
V=UVU^{\dagger}e^{i \epsilon^2}
\]
\[
e^{i\epsilon p}= e^{i \epsilon (U p U^{\dagger} + \epsilon)}
\]
\[
1+ i \epsilon p +{\epsilon^2 \over 2}p^2+ \cdots = 1 + i \epsilon (p+
\epsilon (i  [q,p] +1)  ) + {\epsilon^2 \over 2}(p^2 + \cdots) 
(p^2+ \cdots
\]
In this case $\epsilon$ is not continuous but can be made arbitrarily small.
The operators $p$ and $q$ also depend on the choice of $\epsilon$ which is
reflected in their eigenvalues.  Comparing both sides of this expression we
expect that the usual canonical commutation relations will hold up to
corrections of order $\epsilon$.  To show this note that
given the eigenvalues the operator can be
expressed in a spectral expansion which in the
discrete Weyl representation has the form
\beq
p = \sum_{n=-K}^K n \epsilon
\vert v_n \rangle \langle v_n \vert = {\epsilon^2 \over 2 \pi}\sum_{m,n=-K}^K n \epsilon V^m
e^{{-2 \pi i mn \over M}}= {\epsilon^2 \over 2 \pi}
\sum_{m,n=-K}^K n \epsilon V^m e^{{-i
\epsilon^2 mn }}
\eeq
and
\beq
q = \sum_{k=-K}^K k \epsilon \vert
u_k \rangle \langle u_k \vert = {\epsilon^2 \over 2 \pi} \sum_{l,k=-K}^K k \epsilon U^l e^{{-2
\pi i lk \over M}} = {\epsilon^2 \over 2 \pi}
\sum_{l,k=-K}^K k \epsilon U^l e^{{-i
\epsilon^2 lk }}
\eeq
The commutator can be expressed in this
representation as
\beq
[q,p]= ({\epsilon^2 \over 2 \pi})^2\sum_{klmn=-K}^K n k \epsilon^2 e^{-i
\epsilon^2 (nm+kl)} \left ( U^l V^m - V^m U^l \right )
\eeq
Using
\beq
VU = UVe^{i \epsilon^2}
\eeq
in the above gives
\beq
[q,p]=
({\epsilon^2 \over 2 \pi})^2\sum_{klmn=-K}^K n k \epsilon^2 e^{-i \epsilon^2 (nm+kl)} U^l V^m(1 -
e^{i l m \epsilon^2} ).
\eeq
In order to simplify this note that the
$n$ and $k$ sums can be computed in closed form.  There are 2 cases - if
$m=0$ the sum is $0$ while for $m\not=0$ 
\beq
\sum_{n=-K}^K n
e^{-i \epsilon^2 nm} = e^{i \epsilon^2 Km}\sum_{n=-K}^K n e^{-i
  \epsilon^2 m(n+K)}
\eeq
Next let $n'=n+K$, $n':0\to 2K = 0\to M-1$,
$n=n'-K$:
\beq
\sum_{n'=-K}^K n' e^{-i \epsilon^2 n'm} = e^{-i 
K\alpha}\sum_{n'=0}^{M-1} (n'-K) e^{-i \epsilon^2 mn'} .
\eeq
Let $\alpha :=-\epsilon^2 m$. Then the above can be written as
\beq
\sum_{n=-K}^K n e^{-i \epsilon^2 nm} =
e^{-i 
K\alpha}
\sum_{n=0}^{M-1} (-i {d \over d\alpha}-K) e^{i \alpha n} =
\eeq
\beq
e^{-i 
K\alpha}
(-i {d \over d\alpha}-K) { 1 -e^{i M\alpha } \over
1 -e^{i \alpha }} =
\eeq
\beq
e^{-i 
K\alpha}[ 
{-K (1-e^{iM\alpha}) \over 1-e^{i \alpha}}-
{M e^{iM\alpha} \over 1-e^{i \alpha}} + (-i)(-)(-)(i)
{e^{i\alpha} - e^{i(M+1)\alpha} \over (1-e^{i \alpha})^2}] .
\eeq
Since $e^{iM\alpha} = e^{-2 \pi i m}=1$ this becomes
\beq
-e^{-i 
K\alpha}
{M \over 1-e^{i \alpha}} =
-{2 \pi \over \epsilon^2}
{1 \over 1-e^{-im \epsilon^2}}e^{i \epsilon^2 Km}
\eeq
Using this in the spectral expansion of the commutator gives
\[
[q,p]= ({\epsilon^2 \over 2 \pi})^2
\sum_{klmn=-K}^K n k \epsilon^2 e^{-i \epsilon^2 (nm+kl)} U^l V^m(1 -
e^{i l m \epsilon^2} )=
\]
\beq
({\epsilon^2 \over 2 \pi})^2
\sum_{ml=-K}^Ke^{i \epsilon^2 K(m+l)} \epsilon^2 ({2 \pi \over \epsilon^2})^2  U^l V^m{( 1 -
e^{i l m \epsilon^2}) e^{i \epsilon^2 (M-1)}\over
( 1-e^{-im \epsilon^2})(1-e^{-il \epsilon^2})}=
\eeq
\beq
({\epsilon^2 \over 2 \pi})^2 
\sum_{ml=-K}^K e^{i \epsilon^2 K(m+l)}\epsilon^2 ({2 \pi \over \epsilon^2})^2  U^l V^m{( 1 -
e^{i l m \epsilon^2}) e^{-i \epsilon^2 }\over
( 1-e^{-im \epsilon^2})(1-e^{-il \epsilon^2})} =
\eeq
\beq
\sum_{ml=-K}^K
e^{i \epsilon^2 K(m+l)}
U^l V^m{\epsilon^2( 1 -
e^{i l m \epsilon^2}) e^{-i \epsilon^2 }\over
( 1-e^{-im \epsilon^2})(1-e^{-il \epsilon^2})}
\eeq
Up to this point everything has been exact.
For small $\epsilon$
\[
{( 1 -
e^{i l m \epsilon^2}) e^{-i \epsilon^2 }\over
( 1-\epsilon^2e^{-im \epsilon^2})(1-e^{-il \epsilon^2})} \to
{-i l m \epsilon^2 \over (im \epsilon^2)( il \epsilon^2)}=
{i\over \epsilon^2 }
\]
This means the that leading term in () is
\[
({\epsilon^2 \over 2 \pi})^2({i \over \epsilon^2})\epsilon^2
\sum_{ml=-K}^K  {(2 \pi)^2 \over \epsilon^2}  U^l V^m =
i (\sum_{l=-K}^K U^l)( \sum_{m=-K}^K V^m) =
\]
\[
i \vert u_0 \rangle \langle u_0 \vert v_0 \rangle \langle v_0 \vert
=
i\sqrt{M}  \vert u_0 \rangle \langle v_0 \vert
\]


this becomes

In the
continuum limit the Hilbert space consists of square integrable functions.
We can fix the phase by defining


From these expressions it is possible to investigate 

These wave packets are needed in the coordinate basis in the path 
integral.  The Fourier transform of (\ref{fs.3}) can be computed 
analytically and expressed in terms of the momentum or coordinate uncertainty
of the wave packet:
\beq
\langle x \vert \psi_{i0} (t) \rangle =
(2 \pi)^{-1/2}
\sqrt{{1  \over 1  + i {2 (\Delta p)^2 t \over  \mu}}}
e^{-
{(\Delta p)^2 \over 1  + {4 (\Delta p)^4 t^2 \over  \mu^2}}
(x-{p_it\over \mu})^2}
e^{i{1 \over 1  + {4 (\Delta p)^4 t^2 \over  \mu^2}}
(x p_i   
+ 2 x^2 (\Delta p)^4 {t \over  \mu}
- { t \over  \mu}{p_i^2 \over 2 })} =
\label{fs.4}
\eeq
\beq
(2 \pi)^{-1/2}
\sqrt{{1  \over 1  + i { t \over 2 (\Delta x)^2  \mu}}}
e^{-{1 \over 4 \Delta x^2}
{1 \over 1  + { t^2 \over 4 \Delta x^4   \mu^2}}
(x-{p_it\over \mu})^2}
e^{i{1 \over 1  + {t^2 \over 4 \Delta x^4  \mu^2}}
(x p_i   
+ { x^2 t \over  8 \Delta x^4 \mu}
- { t \over  \mu}{p_i^2 \over 2 })} .
\label{fs.5}
\eeq
where $\Delta p \Delta x = {1 \over 2}$,

What is needed is the discrete analog of a minimal uncertainty state.



Under free time evolution the
center of each of the wave packets moves with the mean momentum and
the size of the wave packet to spreads.


The noninteracting Schr\"odinger equationcan be used to
evolve the initial state from the time of collision
back to a time when the support of the projectile and target
wave packets are be beyond
the range of the
interaction.  Call this time $t=-T$.  Similarly the bob-interacting
Schr\"odinger equation can be used to evolve the final state

evolved forward in time eventually the support of the final wave
packet will be outside of the range of the interaction.  If they are
evolved back to the collision time $t=0$ using the full Hamiltonian we
get a probability amplitude for a transition from wave packets wave
packets that look like free particles in the past and future, but
inclued the effects of the interaction.  The basic elements are
\[
\langle \psi_{f0}(0) \vert U_0^{\dagger}(T)U^{\dagger}(-T) U(T)U_0(-T)\psi_{i0}(0) \rangle =
\]
\[
\langle \psi_{f0}(0) \vert U_0(-T)U(2T)U_0(-T)\psi_{i0}(0) \rangle 
\]
In this case $T$ is finite, and the inital and final states
are coherent states.  The operator
\beq
S(T) : =U_0^{\dagger}(T)U^{\dagger}(-T) U(T)U_0(-T) = e^{2i\delta (T)}
\eeq
is exactly unitary.

There is nothing about this picture that requires a continuum theory,
however attention must be paid to $T$, the range of the potential
the relevant volume, and the structure of the wave packet.

Eventually one would like to get to a sharp momentum scattering amplitude.
While sharp momentum states are normalizable in the discrete representation,
they cannot be used in the above analysis because they are complementary
to the spatial degrees of freedom, which means that they are completely
de-localized in space. The problem in the discrete case is how to
construct the discrete analog of coherent states.  Note as more degrees of
freedom are added, the coherent states representing a given experiment
will have to evolve appropriately.

For this reason it is useful to consider the problem of constructing
minimal uncertainty states in the discrete representation.
The starting point is to define the expectation value and
uncertainty for the discrete position and momentum

These are standard definitions.  Just like in the continuum case the
Schwarz inequality gives
\beq
(\Delta q)^2(\Delta p)^2 =
\Vert
(q-\langle q \rangle) \vert \psi \rangle \Vert^2
\Vert
(p-\langle p \rangle) \vert \psi \rangle \Vert^2 \leq
\vert \langle \psi \vert (q-\langle q \rangle)(p-\langle p \rangle)
\vert \psi \rangle \vert^2
\eeq
Expressing the product of the operators on the right side of (\ref{})
as half the sum of the anti-commutator and commutator gives 
\[
(\Delta q)^2(\Delta p)^2 \geq
\vert \langle \psi \vert
{1 \over 2}\{(q-\langle q \rangle), (p-\langle p \rangle)\}+
{1 \over 2}[(q-\langle q \rangle),(q-\langle q \rangle)]
\vert \psi \rangle \vert^2
\]
The anti-commutator term is real while the commutator term is imaginary
so this becomes
\[
\vert \langle \psi \vert
{1 \over 2}\{(q-\langle q \rangle), (p-\langle p \rangle)\}+
{1 \over 2}[(q-\langle q \rangle),(p-\langle p \rangle)]
\vert \psi \rangle \vert^2 =
\]
\[
\vert \langle \psi \vert
{1 \over 2}\{(q-\langle q \rangle), (p-\langle p \rangle)\}\vert \psi \rangle \vert^2 +
\]
\beq
\vert \langle \psi \vert      
{1 \over 2}[q,p]
\vert \psi \rangle \vert^2 
\eeq

One difference with the continuum case is that the commutator of the discrete
position and momentum operators, 
$[q,p]$, is not constant
\[
[q,p]= [{1 \over M} \sum_{n,m=-K}^K ({U \over u_m})^n m\epsilon,  
{1 \over M} \sum_{k,l=-K}^K ({V \over v_l})^k l\epsilon ]=
\]
\[
[{1 \over M^2} \sum_{mnkl} (U^nV^k e^{{-2 \pi i (mn+kl)\over M}}
(1 - e^{{2 \pi i nk\over M}})ml\epsilon^2 
\]
This vanishes for $n$ or $k=0$.  This can be computed if
$n\not=0$  
note
\[
\sum_{m=-K}^K m e^{-2 \pi i mn/M} = 
e^{2 \pi i Kn/M}
\sum_{m=-K}^K m e^{-2 \pi i (m+K)n/M} =    
e^{2 \pi i Kn/M} \sum_{m=0}^{2K} (m-K) e^{-2 \pi i mn/M}
\]
let $\alpha = -2 \pi n/M$ then this has the form
\[
e^{2 \pi i Kn/M} \sum_{m=0}^{2K} (-i {d \over d\alpha}-K) e^{i\alpha m} =
e^{2 \pi i Kn/M} (-i {d \over d\alpha}-K) 
{ 1- e^{iM\alpha} \over 1- e^{i\alpha }} =
\]
\[
e^{2 \pi i Kn/M} \left ( 
-K{1- e^{iM\alpha}  \over 1- e^{i\alpha}} -
{M e^{iM\alpha} \over 1- e^{i\alpha }}+
{e^{i\alpha}- e^{i(M+1)\alpha} \over (1- e^{i\alpha })^2}\right )= 
\]
For any $n$ $e^{iM\alpha}=1$, however for $n=0$ the denominator vanishes
so it is necessary to return to the original expression to treat
the case that $n=0$.  It can be seen from () and () that $n=0$ does not
contribute.  It follows that () becomes
\[
- e^{2 \pi i Kn/M} {M \over 1-e^{i \alpha}}  \qquad n\not=0
\]

%{(M-K)(1 - e^{i\alpha}) + e^{i\alpha} \over (1- e^{i\alpha })^2}\right )= 
%\]
%\[
%e^{2 \pi i Kn/M} \left ( 
%{(K+1)(1 - e^{i\alpha}) + e^{i\alpha} \over (1- e^{i\alpha })^2}\right )= 
%\]
%\[
%e^{2 \pi i Kn/M} \left ( 
%{1+ K(1 - e^{i\alpha}) \over (1- e^{i\alpha })^2}\right )= 
%\]
%\[
%e^{2 \pi i Kn/M} \left ( 
%{1+ K(1 - e^{{-2 \pi i n \over M}}) \over (1- e^{{-2 \pi i n \over M}})^2 } \right )
%\]
Using this in the expression for the commutator gives
\[=
{\epsilon^2 \over M^2}\sum_{n,k=-K}^K U^n V^k (1 - e^{{2 \pi i n k\over M}})
e^{2 \pi i K(n+k)/M} {M^2 \over  (1-e^{-2 \pi i n \over M})(1-e^{-2 \pi i k \over M})}=
\]
\[
\sum_{n,k=-K}^K U^n V^k 
e^{2 \pi i K(n+k)/M} {\epsilon^2 (1 - e^{{2 \pi i n k\over M}}) \over  (1-e^{-2 \pi i n \over M})(1-e^{-2 \pi i k \over M})}
\]
\[
\sum_{n,k=-K}^K U^n V^k 
e^{\pi i (n+k)/(1+1/2K)} {\epsilon^2 (1 - e^{{2 \pi i n k\over M}}) \over  (1-e^{-2 \pi i n \over M})(1-e^{-2 \pi i k \over M})}
\]
Expanding to leading order in $\epsilon$ gives
\[
\to \sum_{n,k=-K}^K U^n V^k (-1)^{n+k} {-\epsilon^4 nk \over
\epsilon^4 nk}= 
\]
\[
\sum_{n=-K,n \not=0}^K (-1)^nU^n \sum_{k=-K,k \not=0}^K (-1)^k V^k 
\]
\[
\sum_{n=1}^K (-1)^n(U^n+ U^{-n}) \sum_{k=1}^K (-1)^k (V^k+ V^{-k}) 
\]
%\[
%e^{\pi i (n+k)/(1+1/2K)} {\epsilon^2 (1 - e^{{2 \pi i n k\over M}})
%\over  (1-e^{-2 \pi i n \over M})(1-e^{-2 \pi i k \over M})}
%\]
%\[
%\sum_{n,k=-K}^K U^n V^k 
%e^{2 \pi i K(n+k)/M} {\epsilon^2 (1 - e^{{2 \pi i n k\over M}})
%\over  (1-e^{-2 \pi i n \over M})(1-e^{-2 \pi i k \over M})}
%\]
%\[
%\left ( 
%{1+ K(1 - e^{{-2 \pi i n \over M}})
%\over (1- e^{{-2 \pi i n \over M}} )^2}\right )\left ( 
%{1+ K(1 - e^{{-2 \pi i k \over M}})
%\over (1- e^{{-2 \pi i k \over M}} )^2}\right ).
%\]
%Assuming that $K \gg 1$.

On the other hand there are states that make the anti-commutator term
vanish.  Assume a vector $\vert \psi \rangle$ satisfying
\[
(q-\langle q \rangle) \vert \psi \rangle =
i \gamma (p-\langle p \rangle) \vert \psi \rangle
\]
can be found.  It follows that
\[
\vert \langle \psi \vert
{1 \over 2}\{(q-\langle q \rangle), (p-\langle p \rangle)\}
\vert \psi \rangle \vert^2 =
\]
\[
\vert \langle \psi \vert
{1 \over 2} (q-\langle q \rangle)^2{(-i+i)\over \gamma}
\vert \psi \rangle \vert^2 = 0
\]
Given such a solution then the commutator term becomes
\[
{(\Delta p)^4 \gamma^2  } =
{(\Delta q)^4 \over \gamma^2}
\]
or
\[
\gamma = {(\Delta q) \over (\Delta p)}
\]
In this case the commutator makes the inequality into an equal sign.

In the continuum case the equation 
\[
(q-\langle q \rangle) \vert \psi \rangle =
i \gamma (p-\langle p \rangle) \vert \psi \rangle
\]
can be integrated to find the minimal uncertainty state with mean momentum
$\langle p \rangle$ and mean displacement $\langle q \rangle$.  The discrete
version of this equation is
\[
(\epsilon n - \langle q \rangle )\langle u_n \vert \psi \rangle = -i
\gamma \langle p \rangle \langle u_n \vert \psi \rangle + i \gamma
\sum_{k=-K}^K \langle u_n \vert v_k \rangle \epsilon k \langle v_k \vert u_m \rangle
\langle u_m \vert \psi \rangle
\]
or
\[
{1 \over \gamma} (\epsilon n - \langle q \rangle
)\langle u_n \vert \psi \rangle =
\sum_{m=-K}^K
(-i  \langle p \rangle \delta_{mn} +
\sum_{k=-K}^K {i \over M}  \epsilon k
e^{2 \pi i k(n-m)/M} )
\langle u_m \vert \psi \rangle
\]
using () gives
\[
{1 \over \gamma} (\epsilon n - \langle q \rangle
)\langle u_n \vert \psi \rangle =
\sum_{m=-K}^K
(-i  \langle p \rangle \delta_{mn} +
{i \epsilon\over M} e^{-2 \pi i K(n-m) \over M}
{M \over 1-e^{-2\pi i (n-m)/M}})
\langle u_m \vert \psi \rangle =
\]
\[
{1 \over \gamma} (\epsilon n - \langle q \rangle
)\langle u_n \vert \psi \rangle =
\sum_{m=-K}^K
(-i  \langle p \rangle \delta_{mn} +
\epsilon {i e^{{-2 \pi i K(n-m)\over M}} \over 1-e^{-2\pi i (n-m)/M}})
\langle u_m \vert \psi \rangle 
\]



In this case we can do the $k$ sum using () 
\[
\sum_{k=-K}^K \epsilon k
e^{2 \pi i k(n-m)/M} =
\epsilon e^{2 \pi i K (m-n)/M}\left ({1 + K(1 - e^{-2 \pi i (n-m)/M})
\over (1 - e^{-2 \pi i (n-m)/M})^2} \right ) 
\]
which gives
\[
(\epsilon n - \langle q \rangle +  i
\gamma \langle p \rangle 
)\langle u_n \vert \psi \rangle =  i {\gamma \over M} \sum_{m=-K}^K
\epsilon e^{2 \pi i K (m-n)/M}\left ({1 + K(1 - e^{-2 \pi i (n-m)/M})
\over (1 - e^{-2 \pi i (n-m)/M})^2} \right ) \langle u_m \vert \psi \rangle
\]
There remains the question of how to solve this - it is a linear
homogeneous system - if we treat $\gamma$ as an unknown generalized
eigenvalue the it follows that there are $M$ allowed values of $\gamma$
that make the anticommutator term vanish.  Unlike the continuum case
only certain discrete ratios of $\Delta p$ and $\Delta q$ make
anticommutator vanish.  Not that $\gamma$ must be real for
this term to vanish

it is not clear if it has solutions for any
values of $\gamma$, $\langle p \rangle$ or $\langle q \rangle$?



\end{document}

\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{polyzou-fig1.eps}
\caption{\bf V(x),$\langle x \vert \psi_{0i}(0) \rangle$}
\label{fig:1}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{polyzou-fig2.eps}
\caption{\bf V(x),$\langle x \vert \psi_{0i}(-3) \rangle$}
\label{fig:2}
\end{minipage}
\end{figure}


\end{document}

\section{phase shifts}

\section{continuum limit}



\section{to qbits}






intervals $\Delta \phi = \sqrt{{2 \pi \over N}}$

volume  $[-\sqrt{\pi N \over 2} , \sqrt{\pi N \over 2}]$
  
field amplitude and Hilbert space - the goal is to have a well defined
limit as $N\to \infty$

representation of the Hilbert space,  annihilation operators,
momentum space, coordinate space.

formulation of scattering - need unitary time evolution with and
without interaction.

bogoliubov - adiabaticlly turn on interactions

kinematics

conjugate vaariables

we have an infinite number of degrees of freedom in a finite volume

Representation of the Hamiltonain


Free evolution
\[
U(t) = \sum_n V(q) 
\]  
\end{itemize}

\section{Weyl Relations}

Assume $p_i$ and $q_i$ are operators satisfying canonical commutation
relations
\beq
[p_i,q_j] = i \delta_{ij} \qquad [q_i,q-_j] = [p_i,p_j] =0 
\eeq
Define
\beq
U(\mathbf{a}) = e^{i \mathbf{q} \cdot \mathbf{a}}
V(\mathbf{v}) = e^{i \mathbf{p} \cdot \mathbf{b}}
\eeq
Then it follows that
\beq
V(\mathbf{b}) U(\mathbf{a}) = U(\mathbf{a}) V(\mathbf{b}) e^{i \mathbf{a}
\cdot \mathbf{b}}
\eeq
The operators $U(\mathbf{a}) V(\mathbf{b})$ are irreducible in the
sense any operator admits a representation of the form
\beq
O = \sum_{mn} Q_{mn} U(\mathbf{a}_n) V(\mathbf{b}_m)
\eeq
where the sum could be replace by an integral.  The $q_i$ and $p_i$ operators
unbounded while $U(\mathbf{a}) V(\mathbf{b})$ are bounded.

Note that
\[
V(\mathbf{b}_m) U(\mathbf{a}_n) - U(\mathbf{a}_n) V(\mathbf{b}_m)  =
(1 - e^{i \mathbf{a}}
\cdot \mathbf{b}})V(\mathbf{b}_m) U(\mathbf{a}_n) 
\]

The unitary time evolution operator can be expressed in this representation.
It $O$ is unitary then
\[
I =OO^{\dagger} =
\sum Q_{mn} O^*_{kl} U(\mathbf{a}_m) V(\mathbf{b}_n)
V(-\mathbf{b}_l) U(-\mathbf{a}_k) 
=
\sum Q_{mn} O^*_{kl} U(\mathbf{a}_m-a_k) V((\mathbf{b}_n- \mathbf{b}_l)
e^{-i\mathbf{a}_k \cdot (\mathbf{b}_n- \mathbf{b}_l)}

\int O(p,q)O^*(p',q') e^{-i p\cdot (q-q')} =
\delta (p-p') \delta (q-q')

The continuum case is simpler




Projection operators:

\[
\langle x_k \rangle \langle x_k \vert =
\sum_{n=1}^N{1 \over 2^N} \left ( {X\over x_k}\right )^n 
\]
\[
\langle z_k \rangle \langle z_k \vert =
\sum_{n=1}^N{1 \over 2^N} \left ( {X\over z_k}\right )^n 
\]

The unitary operators $X$ and $Z$ can be used to construct states
\[
\vert Z \rangle = {1 \over \sqrt{2^N}}
X^n \vert z_m \rangle
\]
or
\[
\vert X \rangle = {1 \over \sqrt{2^N}}
Z^n \vert z_m \rangle
\]
these states are independent of $m$ and seem closely related to the
states that you talked about?

\vfill\eject

{\bf Basic building blocks of quantum logic -  Q-bits}

\bigskip
\[
{\cal H} =\mbox{ 2 dimensional complex vector space}
\]

\vfill\eject

{\bf Higher dimensions:}

\bigskip

Let ${\cal H}$ be a $2^N$ dimensional Hilbert space.  Let $A$ be any Hermitian operator
on ${\cal H}$.  Then $A$ has $2^N$ eigenvalues, $a_n$, and an orthonormal basis
of eigenvectors $\{ \vert a_n \rangle \}$.

Define a unitary ``translation'' operator $X$ (generalize $\sigma_1$):
\[
odX\vert a_n \rangle = \vert a_{n+1} \rangle
\qquad
X\vert a_{2^N} \rangle = \vert a_{1} \rangle
\]
Since $(X^{2^N}-1)\vert a_n \rangle =0$ for any basis vector $\vert a_n \rangle$,
$X$ has eigenvalues $x_n = 2^{2\pi i n \over 2^N}$ on the unit circle
and orthonormal eigenvectors $\vert x_n \rangle$.  
The arbitrary phases of the eigenvectors, $\vert x_k \rangle$, can be chosen
so
\[
\langle a_m \vert x_k \rangle ={1 \over \sqrt{2^N}} e^{-{2\pi i kn \over 2^N}}.
\]
Define another translation operator $Z$ so
\[ 
\langle x_k \vert Z = \langle x_{k+1} \vert
\qquad
\langle x_{2^N} \vert Z = \langle x_{1} \vert
\qquad
\langle x_k \vert (Z^{2^N}-1) = 0.
\]
$Z$ also has eigenvalues $z_n = 2^{2\pi i n \over 2^N}$ and orthonormal
eigenvectors $\{\vert z_n \rangle \}$.  The arbitrary phases
of the $\vert z_n \rangle$ can be chosen so
\[
\langle z_m \vert x_k \rangle ={1 \over \sqrt{2^N}} e^{-{2\pi i kn \over 2^N}}
\]
which gives $\vert z_k \rangle = \vert a_k \rangle$.

\bigskip

The operators $W$ and $Z$ are irreducible in the sense that any operator
on ${\cal H}$ is a polynomial of the form
\[
A = \sum_{m,n=1}^{2^N} a_{mn} X^m Z^n.
\]
$X$ and $Z$ have the property
\[
ZX= X Ze^{i {2 \pi \over 2^N}}
\]
This is a discrete version of the Weyl algebra; it also generalizes
the irreducible Q-bit algebra.

\bigskip



\vfill\eject

{\bf Connection with Q-bits}

\bigskip

Note $X$ and $Z$ are discrete unitary translation operators on the
unit circle of $2^N$ equally spaced eigenvalues.

\bigskip

Since $2^N$ has $N$ prime factors of $2$ we can replace the $2^N$ indices
$m$ by $N$ indices $(n_1 \cdots n_N)$, 
$x_m = x_{n_1 \cdots n_N}$, where each of the $n_i$ have 2 values.  For example
for $N=2$ the $m$ and $(n_1,n_2)$ are related by 
\[
1 \leftrightarrow (0,0) \qquad
2 \leftrightarrow (1,0) \qquad
3 \leftrightarrow (0,1) \qquad
4 \leftrightarrow (1,1)
\]
Then powers of
$X$ and $Z$ can expressed as products of new operators $X_i$ and $Z_i$ operator that
act on the $i^{th}$ index like $\sigma_1$ and $\sigma_3$:
\[
X_1 (0,0) = (1,0) \qquad X_2 (0,0) = (0,1)
\]
\[
X_1(1,0) = (0,0) \qquad X_2 (0,1) = (0,0)
\]
\[
X_1 (0,1) = (1,1)  \qquad X_2 (1,0) = (1,1)
\]
\[
X_1(1,1) = (0,1)  \qquad X_2 (1,0) = (1,1)
\]
etc.

Doing the same with $Z_n$ 
gives an irreducible set of operators on the $2^N$ dimensional
Hilbert space expressed in terms of products of gates $\sigma_1$ and
$\sigma_3$ that act on products of $N$ q-bits.  On this space an
arbitrary linear operator $A$ has the represetation:
\bigskip
\[
A = \sum_{mn} b_{mn} X^m Z^n =
\sum a_{m_1 \cdots m_N,n_1 \cdots n_N} \prod_{k,l=1}^N X_k^{m_k} Z_l^{n_l}.
\]
\[
X^m = X_1^{n_1} X_2^{n_2} \cdots X_N^{n_N} \qquad
Z^m = Z_1^{n_1} Z_2^{n_2} \cdots Z_N^{n_N}
\]
\[
X_i Z_i = Z_i X_i e^{{2\pi i \over 2}}
\]
\[
[X_i, W_j]= [X_i, Z_j]=[Z_i,Z_j]=0 \qquad i\not= j 
\]
\vfill\eject

{\bf Continuum limit}

\bigskip

If $2^N$ is large we can consider an approximation to the continuum
by defining the small quantity $\epsilon$ and real numbers
$p_n$ and $q_n$ by 
\[ 
{2 \pi \over 2^N} = \epsilon^2
\qquad
p_n=q_n = n\epsilon .
\]
spacing between adjacent numbers:
\[
p_{n+1}-p_n = q_{n+1}-q_n = \epsilon
\]
range on numbers:
\[
-{\pi \over \epsilon} \leq p_n,q_n \leq {\pi \over \epsilon} 
\]

We also can define self-adjoint operators $\hat{p}$ and $\hat{q}$ by
\[
X=e^{i \epsilon \hat{q}} \qquad Z=e^{i \epsilon \hat{p}} . 
\]
We define
\[
V(q) := Z^n =  e^{i \hat{p} q_n}  \qquad  U(p) :=  X^n  = e^{i \hat{q} p_n} 
\]
In this notation the relation
\[
Z^lX^k= X^kZ^le^{2 \pi i kl/2^N}
\]
becomes
\[
e^{iq \hat{p}} e^{ip\hat{q}} = e^{ip\hat{q}}e^{iq\hat{p}}e^{ipq}
\]
or
\[
V(q)U(p) = U(p)V(p)e^{ipq} 
\]
which is just the Weyl form of the canonical commutation relations.
While everything is discrete,
%if $[0,2\pi]$ is replaced by
%$[-\pi,\pi]$
the spectrum of $\hat{q}$ and
$\hat{p}$ approach the real line, 
and the spacing between
eigenvalues of $\hat{p}$ and $\hat{q}$
go to 0 as $N$ get large.  To get a $L^2(\mathbb{R})$ representation in the
$\lim_{N \to \infty}$ the states should be 
limited to states that vanish near $\pm \pi$.  In the limit $N\to\infty$
these become the continuum Weyl relations.

\bigskip

The Weyl  operators are an irreducible set of operators, so
they can be used to represent any operator on the space.
\[
A = \sum_{mn}A(p_m,q_m) U(p_m)V(q_n) \epsilon^2  
\to 
\int A(p,q) U(p)V(q) dp dq
\]

\bigskip

This outlined a path from q-bits to continuum quantum mechanics in terms
of coordinate and conjugate momentum variables.

\bigskip

In general multiplication operators can be expressed as
\[
F(\hat{p}) = \sum_{m=1}^{2^N} F(p_m) \sum_{n=1}^{2^N} ({Z \over z_m})^n
\]
\[
F(\hat{q}) = \sum_{m=1}^{2^N} F(q_m) \sum_{n=1}^{2^N} ({X \over x_m})^n
\]


\[
e^{i {\hat{p}^2 \over 2 \mu}t}  = \sum_{m=1}^{2^N}
e^{i {{p}_m^2 \over 2 \mu}t}\sum_{n=1}^{2^N} e^{-2\pi i mn \over 2^N} Z^n 
\]
\[
e^{-i V(\hat{q})t} = \sum_{m=1}^{2^N} e^{-i V(\hat{q}_m)t}
\sum_{n=1}^{2^N} e^{-2\pi i mn \over 2^N} X^n
\]
The product of these operators gives the coefficients of the
approximate transfer matrix
\[
e^{-i {\hat{p}^2 \over 2 \mu}t} e^{-i V(\hat{q})t} =
 \]
\[
\sum_{n=1}^{2^N}
\sum_{l=1}^{2^N}
\left (\sum_{m=1}^{2^N}
\sum_{k=1}^{2^N}
e^{-i ({p_m^2 \over 2 \mu}+V(\hat{q}_m) \Delta t -{2\pi i (mn+lk) \over 2^N}} \right )Z^nX^l =
\]
\[
\sum_{n=1}^{2^N}
\sum_{l=1}^{2^N}
T_{nl}Z^nX^l
\]
Squaring this gives

\[
T^2=
\sum_{n=1}^{2^N}
\sum_{l=1}^{2^N}
T_{n_1l_1}T_{n_2l_2}e^{-{2 \pi i l_1n_2 \over  2^N}} Z^{n_1+n_2}X^{l_1+l_2}
\]

{\bf Discrete representation}

Products of operators:
\[
X^{m_1} Z^{k_1} X^{m_2} Z^{k_2} = X^{m_1+m_2}Z^{k_1+k_2}
e^{{2 \pi i k_1m_2 \over  2^N}}
\]
\[
p= n\epsilon \qquad
q=n\epsilon
\]
\[
V(n\epsilon)U(m\epsilon) = U(m\epsilon)V(n\epsilon)e^{i{mn 2\pi \over 2^N}} 
\]

Motivated by work by P. Muldowney, P. J{\o}rgensen, E. Nathanson, and J. Schwinger.

\bigskip

References:

\bigskip

J. Schwinger, Quantum Mechanics, (Springer-Verlag, Berlin, Heidelberg, 2001).

\bigskip

P. Muldowney, A Modern Theory of Random Variation (Wiley, N.J., 2012).

\bigskip

E. S. Nathanson and E. Jørgensen, Palle, J. Math. Phys.56, 092102 (2015).

\bigskip

E. S. Nathanson, University of Iowa Thesis (2015).

\bigskip

W. N. Polyzou and Ekaterina Nathanson,  Phys. Rev. C 101,064001(2020). 

   
\vfill\eject

In term of   
\vfill\eject

{\bf Formulation of the scattering problem in terms of canonical variables:}

\bigskip

For a Hamiltonian of the form $H=p^2/2\mu + V(q)$, with a short range potential
$V(q)$, the probability of a particle initially heading towards the scattering
center to be detected in a detector is
\[
P_{fi} = \vert \langle f\vert i \rangle \vert^2
\]
where $\vert i \rangle$ represents the state of the incident particle and target at
a given time and $\vert f \rangle$ represents the state measured in the
detector at a common time.  The result is independent of the common time since
time evolution is unitary, but the scattering probability  cannot be
evaluated using initial and final states
at different times.

These states have to be evaluated at the same time, but there is no
time when they are simple.  One way to calculate them at a common time
is to use initial conditions at large positive or negative times, where the states are
simple, and evolve both back to $t=0$.  In this case the initial
conditions are replaced by the scattering asymptotic condition
\[
\lim_{t \to  \infty }\Vert e^{-iHt}\vert f\rangle  - e^{-i{p^2\over 2\mu}t}
\vert f_0 \rangle \Vert =0
\qquad
\lim_{t \to  -\infty }\Vert e^{-iHt}\vert i\rangle  - e^{-i{p^2\over 2\mu}t}
\vert i_0 \rangle \Vert =0
\]
where $\vert f_0 \rangle$ and  $\vert i_0 \rangle$ are the states that
would be seen at time $0$ in the absence of interactions.
The unitarity gives
\[
\lim_{t \to  \infty }\Vert   \vert f\rangle  - e^{iHt}e^{-i{p^2\over 2\mu}t}
\vert f_0 \rangle \Vert =0
\qquad
\lim_{t \to  -\infty }\Vert\vert i\rangle  -e^{iHt} e^{-i{p^2\over 2\mu}t}
\vert i_0 \rangle \Vert =0
\]
where $\vert i_0 \rangle$ and  $\vert f_0 \rangle$ are localized
free particle states
at time 0 with mean momenta of the physical time 0 states.
We define:
\[
\Omega_{\pm}(t) : =e^{\pm iHt}e^{\mp i{p^2\over 2m}t}
\]
which for sufficiently large $t=T$ maps 
$\vert i_0 \rangle$ and  $\vert f_0 \rangle$ to
$\vert i \rangle$  and $\vert f \rangle$ respectively.
In this way the scattering probability can be expressed as
\[
P_{fi} = \vert \langle f\vert i \rangle \vert^2 \approx
\vert \langle f_0\vert \Omega_+^{\dagger} (T)
\Omega_-(T)\vert  i_0 \rangle \vert^2
\]
The operator
\[
S:= \lim_{T\to \infty} \Omega_+^{\dagger} (T)
\Omega_-(T)
\]
is the scattering operator.  A formal treatment of the $T$-limit allows this
to be expressed as
\[
S \approx  I - 2 \pi i \delta (E_f -E_i) \langle p_f \vert
V \Omega_-(T) \vert p_i \rangle
\]
for sufficiently large $T$ (this is actually an integral kernel).
Here $p_i$ and $p_f$ are the initial and final momenta.
The main dynamical content is in 
\[
\langle p_f \vert
V \Omega_-(T) \vert p_i \rangle =
\langle p_f \vert
V e^{-iHT} e^{i {p_i^2 \over 2m}T} \vert p_i \rangle=
\langle p_f \vert
V e^{-iHT} \vert p_i(-T) \rangle
\]
where the initial and final sharp momentum states are replaced by 
Gaussian wave packets with a delta function normalization
\[
\int \langle p \vert f_0 \rangle  dp =1
\qquad
\int \langle p \vert i_0 \rangle  dp =1 .
\]
because the asymptotic conditions involve strong limits.
Standard methods of scattering theory reduce this to
the computation of the following quantity
\[
\langle k_f \vert T \vert k_i \rangle \approx 
\langle f_0 \vert  V e^{-iHT}
U_0(-T)\vert i_0  \rangle
\]
where
\[
U_0(t)=  e^{-i{p^2 \over 2\mu}t}
\]
and the initial and final free particle states are normalized so they
integrate to 1.  This formulation has the advantages
\begin{itemize}
\item[1)] Wave packet spreading in the continuous spectrum is limited to the times
  when the state is within the range of the potential due to $U_0(-T)$.  This is
  because $\Omega (T) \to 1$ when the particle is outside of the range of the
  potential.

\item[2)] The relevant volume is finite - it limited by the range of
the potential and the spatial uncertainty of the wave packet.  The effect of
$\Omega(T)$ is to transform the non-interacting Gaussian wave packet
at time 0 to the corresponding interacting one at time 0.

\item[3)] The relevant value of $T$ is finite - it must be large
enough for the ``particle'' to be out of the range of the potential.
The final factor of $V$ provides an additional volume cutoff.  
\end{itemize}

Some issues: it is desirable to make the momentum and position
uncertainty in the wave packet small - narrow wave packets in momentum
are needed to ensure that the $T$ operator is approximately constant on the
support of the Gaussian function - unfortunately making the momentum
uncertainty small increases the scattering volume.

The dynamical part of the calculation is the computation of
$e^{-iHT}$, but this is what quantum computers are designed to do.
One issue is that $H$ is the sum on non-commuting operators
(this is necessary for scatteirng).

This can be treated using the Trotter product formula which is the
operator version of
\[
\lim_{n \to \infty} (1+x/n)^n = e^x .
\]
For non-commuting operators it implies
\[
e^{A+B} = \lim_{N\to \infty} (e^{A/N}e^{B/N})^N
\]
Applied to the time evolution operator gives
\[
e^{-iHt} = \lim_{N\to \infty} (e^{-i{p^2 \over 2\mu}t/N}e^{-iVt/N})^N
\]
which express this as an infinite product of non-commuting
unitary operators.  We define
\[
\Delta t = T/N
\]
This leads to the ``Feynman path integral''
which involves the computation of
\[
\langle q_f,t \vert e^{iHt} \vert q_i,0\rangle .
\]
After some standard manipulations (integrating out the intermediate momenta)
this has the form
\[
\langle q_f,t \vert e^{iHt} \vert q_i,0\rangle = \int \prod K( q_f, q_N,\Delta t)dx_N e^{-i V(x_N)\Delta t} 
\cdots
K( q_1, q_i,\Delta t)dq_1 e^{-i V(q_1)\Delta t} \delta(q_1-q_i) 
\]
where
\[
K( q_{i+1}, q_i,\Delta t) :=
\langle q_{i+1} \vert e^{i {\mathbf{p}^2 /2\mu}\Delta t}
\vert q_i \rangle =
(-{\mu \over 2 \pi i \Delta t})^{N/2}
e^{
i {\mu \over 2\Delta t} (q_{n-1}-q_{n})^2}  
%}
%\label{njm.6}
\]
The $K( q_{i+1}, q_i,\Delta t)$ have
the property that
\[
\int dq_{i} K( q_{i+1}, q_i,\Delta t) =1
\]
This normalization is due to the unitarity of free propagation.

\bigskip

We interpret
\[
K( q_{i+1}, q_i,\Delta t) dq_{i}
\]
as the complex probability for a particle
arriving at $q_{i+1}$ after time $\Delta t$
if it starts within
$dq_{i}$ of $q_{i}$.
%The normalization condition implies that
%it had to come from one of the intervals.

\bigskip

\vfill\eject

\begin{figure}
\centering
\caption{Plot of the potential and the incident initial free wave packet
at time $t=-3.0$.}
%\includegraphics[scale=.5]{figure_1.pdf}
%\includegraphics[scale=.5]{fig_3.pdf}
\includegraphics[scale=.5]{fig_1.pdf}
%\caption{Potential and initial wave packet at $t=-3$}
%\caption{Plot of the potential and the incident initial free wave packet
%at time $t=-3$.}
\label{figure 1}
\end{figure}  

\vfill\eject

\begin{figure}
\caption{Plots of the delta function normalized Gaussian,
the exact Born approximation to $T$, and the Born approximation
smeared with the initial Gaussian wave packet.}
\centering
%\includegraphics[scale=.5]{fig19.pdf}
%\includegraphics[scale=.5]{fig_20.pdf}
\includegraphics[scale=.5]{fig_11.pdf} 
%\label{figure 23}
%\label{figure 20}
\label{figure 11}
\end{figure}

\vfill\eject

\begin{figure}
\caption{Shift in the phase of the real and imaginary parts of the incident wave packet at the collision time $t=0.0$}
\begin{minipage}[t]{.45\linewidth}
\centering
%\includegraphics[scale=.5]{fig_6.pdf}
\includegraphics[scale=.5]{fig_4a.pdf}
%\caption{Shift in the phase of the real part of the incident wave packet at the% collision time $t=0$}
%\caption{Re($\langle x \vert \psi_{0i}(0) \rangle$)
%and Re($\langle x \vert \Omega(t=-3)\vert \psi_{i0}(0) \rangle$)}
%\label{figure 6}
%\end{figure}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
%\begin{figure}
\centering
%\includegraphics[scale=.5]{fig_7.pdf}
\includegraphics[scale=.5]{fig_4b.pdf}
%\caption{Shift in the phase of the imaginary part of the incident wave packet at the collision time $t=0$}
%\caption{Im($\langle x \vert \psi_{0i}(0) \rangle$)
%and Im($\langle x \vert \Omega(t=-3)\vert\psi_{i0}(0) \rangle$)}
%\label{figure 7}
\end{minipage}
\label{figure 4}
\end{figure}  
\vfill\eject

{\bf Complex probabilities}

\bigskip

In the simplest case probabilities can be defined by assigning non-negative
probabilities to sub-intervals $I$ of some interval, $I_T$, where
$P_I$ is the probability that a measurement will
fall in the interval $I$. Normally probabilities should satisfy the obvious conditions:
\[
0 \leq P_I \leq 1  \qquad P_{I_T}=1
\]
\[
P_I + P_{I^c} =1
\]
\[
P(\cup_{n=1}^\infty I_n )= \sum_{n=1}^\infty P_{I_n}   \qquad I_n \cup I_m = {\empty}
\qquad n\not =m
\]
In this case probabilities are defined on sets that are generated by
intervals using complements and countable unions.  If these are
generated by intervals then the probability is a Lebesgue measurable function
on the $\sigma$-algebra of Lebesgue measurable sets.

\bigskip

Because the value of a countable sum of non-positive or complex
quantities can be finite or infinite, depending on the order in which
the infinite quantities are added, the above definition cannot be
extended to non-positive or complex probabilities.  If countable
additivity is replaced by finite additivity, and we still want
probabilities defined on sets generated by complements and finite
unions, we need a general integral that is defined on finitely
generated measurable sets.  An generalization of the ordinary Riemann
integral, due to Henstock and Kurzwell, is defined on the sets
generated by intervals under complements and finite unions. This
definition is consistent with the standard notion of probability for
positive probabilities, but can be extended to non-positive and
complex probabilities.  This is the interpretation assigned to the
complex probability
\[
dP:= 
dq_{i} K( q_{i+1},t, q_i,\Delta t) .
\]

\vfill\eject

{\bf Cylinder sets and dynamics}

\bigskip

If we multiply $K( q_{i+1}, q_i,\Delta t)$ by $e^{- iV(q_i) \Delta t}$
this is the kernel of {\it unitary} approximation to the {\it transfer matrix}
$e^{-iH\Delta t}$ through a time step $\Delta t$
\[
K( q_{i+1}, q_i,{t\over N})e^{- iV(q_i) \Delta t}=
\langle q_{i+1} \vert e^{-i{p^2\over 2\mu}\Delta t} e^{-i V\Delta t}
\vert q_i\rangle \approx  
\langle q_{i+1} \vert e^{-i H\Delta t} \vert q_i\rangle
\]

This gives
\[
\int \langle q_f \vert e^{-iHt} \vert q_1,0\rangle
d{q_1} \langle q_1 \vert f_0(-T) \rangle
=
\]
\[
\lim_{N \to \infty} \int 
\prod_{i=1}^N (T( q_{i+1}, q_i,{\Delta t})dq_i)  
\langle q_1 \vert f_0(-T) \rangle 
=
\]
\beq
\lim_{N \to \infty} 
\int \prod_{i=1}^N (K( q_{i+1}, q_i,{\Delta t})
e^{-i V(q_i) \Delta t} dq_i)  
\langle q_1 \vert f_0(-T) \rangle . 
\label{a}
\eeq
A spatially localized wave packet unitarily evolved over a finite time
should remain localized (see figure).  This means that the relevant
integration volume $\Lambda$ is finite.  We can decompose $\Lambda$
into a set of small cells, with the property that $V(q)$ is
approximately constant on the $k^{th}$ cell.

The integrals associated with each time slice can be replaced by sums
of integrals over cells:  The result is
\[
\langle p_f \vert
V \Omega_-(T) \vert p_i \rangle =
\langle p_f \vert
V e^{-iHT} e^{i {p_i^2 \over 2\mu}T} \vert p_i \rangle =
\]
\[
\lim_{N \to \infty} 
\sum_{n_1} \int_{I_{n_1}} dq_1 \cdots
\sum_{n_N} \int_{I_{n_N}} dq_N  
\langle p_f \vert
V \vert q_f \rangle dq_f
\prod_{i=1}^N (K( q_{i+1},t, q_i,\Delta t)
e^{-i V(q_i) \Delta t} dq_i)  
\langle q_1 \vert f_0(-T) \rangle = 
\]
Next assume that the cells are sufficiently small so $V(q)$
multiplied by the preceding wave function is approximately constant the cell.
With this approximation this nested set of integrals becomes a sum of products of integrals over small cells [J{\o}rgensen and Nathanson path integral]:

We define
\[
P(q_n ,I_k) := \int_{cell\, i} dq
K( q_{n},t_k, q_k,\Delta t) \approx
K( q_{n},t_k, q_k,\Delta t) dq
\]
which has the property
\[
\sum_{cells} P(q ,I_m) = 1
\]
where $q_{N+1} = q_f$.  More generally integrating over cells for each
time slice gives a complex probability for a path to pass through cell
$I_1$ at time $\Delta t$, $I_2$ at time $2\Delta t$, $\cdots$
$I_N$ at time $N\Delta t$:
\[
\sum_{cells} P(q,I_N,\cdots I_1)=1 
\]
The set of paths passing through $I_1$ at time $\Delta t$, $I_2$ at time $2\Delta t$, $\cdots$
$I_N$ at time $N\Delta t$ defines a cylinder set of paths.

If the potential is approximately locally constant on each cell
it can be evaluated at any point in the cell.  In this case
\[
\langle p_f \vert e^{-iHt} \vert f_0(-T) \rangle \approx
\sum_{cylinder sets} P(q,I_N,\cdots I_1) e^{-i \sum V(q_{i_j})\Delta t}
\langle q_{1_j} \vert f_0(-T) \rangle 
\]
where $q_{i_j}$ is the $j^{th}$ in the $i^{th}$ cell.
Nathanson J{\o}rgensen show this converges to a global solution to the
Schr\"odinger equation.

A direct calculation using this method
is not practical due to the large number of cylinder sets
$M^N$, $M=$ number of intervals / time slice, $N=$ number of time slices.

To get a numerically manageable approximation use the decomposition into
into cylinder set in equation (\ref{a}).
This gives
\[
P(q,I_N,\cdots I_1) \approx
K( q, q_N,\Delta t)dq_N
K( q_{N},t, q_{N-1},\Delta t)dq_{N-1} \cdots
K( q_{2},t, q_{1},\Delta t)
\]

\[
\langle p_f \vert
V \Omega_-(T) \vert p_i \rangle =
\langle p_f \vert
V e^{-iHT} e^{i {p_i^2 \over 2\mu}T} \vert p_i \rangle \approx
\]
\[
\sum_{n_1} dq_1 \cdots
\sum_{n_N}  dq_N  
\langle p_f \vert
V \vert q_f \rangle dq_f
\prod_{i=1}^N (K( q_{i+1}, q_i,{t\over N})
e^{-i V(q_i) \Delta t} dq_i)  
\langle q_1 \vert f_0(T) \rangle  
\]
where the integrals are replaced by sums over cylinder sets.
The computation involves computing the $N^{th}$ (number of time slices)
of the $M \times M$ (number of cells in one time slice) ``transfer'' matrix:
\[
K( q_{i+1}, q_i,\Delta t)
e^{-i V(q_i) {t\over N}} dq_i
\]
which can be done more efficiently.

\bigskip

The product
\[
\prod_{i=1}^N (K( q_{i+1}, q_i,\Delta t)dq_i
\]
represents the approximate complex probability that a path goes through
the cylinder set defined by the cells $I_1$ at time $t_1, \cdots , I_k$ at time $t_k \cdots$.  The cylinder sets 
define an equivalence relation on paths.   Every path is
in one of the equivalence classes, and each class in assigned a
complex probability.

\bigskip

$\bullet$ The potential only needs to be known at one point in each cell ($M$ points).

\bigskip

$\bullet$ While the matrix is unitary, when it is applied to a localized initial
state that is evolved for a finite time, the result will still
be localized in a finite volume.


\bigskip

$\bullet$ While the number of cylinder sets is prohibitively large, this method
treats all of the paths in parallel, which results in a polynomial rather
than exponential growth as the size of the problem gets bigger.

\bigskip

$\bullet$ It is possible to extract the complex probability associated with any
cylinder set.

\bigskip

$\bullet$ In this representation the complex probabilities correspond to free
propagation, while the interactions are the analogs of gates.  The
complication is that in this representation free propagation is not local.

\bigskip

$\bullet$ In principle the conjugate coordinate and momentum variables can
ultimately be expressed in terms of $q$-bits.  Canonical fields
smeared with any orthonormal basis have the structure of a system with
an infinite number of canonical coordinates and momenta.  The path
integral has the same structure, except there are an infinite number
of modes (``particles'').

\bigskip

$\bullet$ The size of the matrix in this work was 5000-20000, which gives some
notion of the scale of a real calculation.

\begin{table}
\caption{Sums of the real and imaginary parts of the complex probabilities.} 
\begin{tabular}{lll}
\hline
\hline
$x$ & $\sum_n Re (P_n(x))$ &$\sum_n Im (P_n(x))$\\
\hline
-25.005001 & p=1.000000$\times 10^0$ & + i 5.273559$\times 10^{-16}$ \\
%-24.604921 p=1.000000e+00 + i(9.992007e-16)
%-24.204841 p=1.000000e+00 + i(5.828671e-16)
%-23.804761 p=1.000000e+00 + i(-1.415534e-15)
%-23.404681 p=1.000000e+00 + i(-1.193490e-15)
%-23.004601 p=1.000000e+00 + i(4.996004e-16)
%-22.604521 p=1.000000e+00 + i(6.383782e-16)
%-22.204441 p=1.000000e+00 + i(-1.387779e-16)
%-21.804361 p=1.000000e+00 + i(1.110223e-16)
%-21.404281 p=1.000000e+00 + i(1.942890e-16)
%-21.004201 p=1.000000e+00 + i(3.330669e-16)
%-20.604121 p=1.000000e+00 + i(-1.332268e-15) 
-20.204041 &p=1.000000$\times 10^0$ &- i 1.387779$\times 10^{-16}$ \\
%-19.803961 p=1.000000e+00 + i(-1.471046e-15)
%-19.403881 p=1.000000e+00 + i(-1.304512e-15)
%-19.003801 p=1.000000e+00 + i(-6.106227e-16)
%-18.603721 p=1.000000e+00 + i(-9.714451e-16)
%-18.203641 p=1.000000e+00 + i(-2.498002e-16)
%-17.803561 p=1.000000e+00 + i(-4.440892e-16)
%-17.403481 p=1.000000e+00 + i(4.440892e-16)
%-17.003401 p=1.000000e+00 + i(-2.775558e-16)
%-16.603321 p=1.000000e+00 + i(3.608225e-16)
%-16.203241 p=1.000000e+00 + i(1.221245e-15)
%-15.803161 p=1.000000e+00 + i(8.604228e-16)
%-15.403081 p=1.000000e+00 + i(2.220446e-16)
-15.003001 & p=1.000000$\times 10^0$ & + i 2.775558$\times 10^{-17}$ \\
%-14.602921 p=1.000000e+00 + i(-9.436896e-16)
%-14.202841 p=1.000000e+00 + i(1.249001e-15)
%-13.802761 p=1.000000e+00 + i(-8.326673e-17)
%-13.402681 p=1.000000e+00 + i(-2.775558e-16)
%-13.002601 p=1.000000e+00 + i(1.665335e-16)
%-12.602521 p=1.000000e+00 + i(2.498002e-16)
%-12.202440 p=1.000000e+00 + i(0.000000e+00)
%-11.802360 p=1.000000e+00 + i(-2.498002e-16)
%-11.402280 p=1.000000e+00 + i(-6.661338e-16)
%-11.002200 p=1.000000e+00 + i(-9.992007e-16)
%-10.602120 p=1.000000e+00 + i(-8.049117e-16)
-10.202040 & p=1.000000$\times 10^0$ &+ i 2.775558$\times 10^{-16}$ \\
%-9.801960  p=1.000000e+00 + i(2.775558e-17)
%-9.401880  p=1.000000e+00 + i(9.714451e-16)
%-9.001800  p=1.000000e+00 + i(1.110223e-15)
%-8.601720  p=1.000000e+00 + i(-6.383782e-16)
%-8.201640  p=1.000000e+00 + i(0.000000e+00)
%-7.801560  p=1.000000e+00 + i(3.608225e-16)
%-7.401480  p=1.000000e+00 + i(1.554312e-15)
%-7.001400  p=1.000000e+00 + i(5.551115e-17)
%-6.601320  p=1.000000e+00 + i(-4.996004e-16)
%-6.201240  p=1.000000e+00 + i(-3.053113e-16)
%-5.801160  p=1.000000e+00 + i(-1.942890e-16)
%-5.401080  p=1.000000e+00 + i(-6.383782e-16) 
-5.001000  & p=1.000000$\times 10^0$ &+ i 7.771561$\times 10^{-16}$ \\
%-4.600920  p=1.000000e+00 + i(8.049117e-16)
%-4.200840  p=1.000000e+00 + i(1.498801e-15)
%-3.800760  p=1.000000e+00 + i(1.110223e-15)
%-3.400680  p=1.000000e+00 + i(1.415534e-15)
%-3.000600  p=1.000000e+00 + i(1.471046e-15)
%-2.600520  p=1.000000e+00 + i(2.498002e-16)
%-2.200440  p=1.000000e+00 + i(-5.551115e-17)
%-1.800360  p=1.000000e+00 + i(5.273559e-16)
%-1.400280  p=1.000000e+00 + i(9.436896e-16)
%-1.000200  p=1.000000e+00 + i(-7.494005e-16)
%-0.600120  p=1.000000e+00 + i(-1.276756e-15)
%-0.200040  p=1.000000e+00 + i(-7.494005e-16)
0.200040   &p=1.000000$\times 10^0$ &+ i 3.608225$\times 10^{-16}$ \\
%0.600120   p=1.000000e+00 + i(-4.996004e-16)
%1.000200   p=1.000000e+00 + i(-3.330669e-16)
%1.400280   p=1.000000e+00 + i(6.938894e-16)
%1.800360   p=1.000000e+00 + i(1.137979e-15)
%2.200440   p=1.000000e+00 + i(1.748601e-15)
%2.600520   p=1.000000e+00 + i(9.992007e-16)
%3.000600   p=1.000000e+00 + i(1.471046e-15)
%3.400680   p=1.000000e+00 + i(-1.110223e-16)
%3.800760   p=1.000000e+00 + i(-2.220446e-16)
%4.200840   p=1.000000e+00 + i(-7.494005e-16)
%4.600920   p=1.000000e+00 + i(1.137979e-15)
5.001000   &p=1.000000$\times 10^0$ &+ i 1.665335$\times 10^{-16}$ \\
%5.401080   p=1.000000e+00 + i(-1.693090e-15)
%5.801160   p=1.000000e+00 + i(-1.443290e-15)
%6.201240   p=1.000000e+00 + i(-3.330669e-16)
%6.601320   p=1.000000e+00 + i(-8.881784e-16)
%7.001400   p=1.000000e+00 + i(2.775558e-16)
%7.401480   p=1.000000e+00 + i(1.415534e-15)
%7.801560   p=1.000000e+00 + i(1.859624e-15)
%8.201640   p=1.000000e+00 + i(6.383782e-16)
%8.601720   p=1.000000e+00 + i(1.165734e-15)
%9.001800   p=1.000000e+00 + i(9.159340e-16)
%9.401880   p=1.000000e+00 + i(2.498002e-16)
%9.801960   p=1.000000e+00 + i(-1.942890e-16)
10.202040  &p=1.000000$\times 10^0$ &+ i 5.273559$\times 10^{-16}$ \\
%10.602120  p=1.000000e+00 + i(3.053113e-16)
%11.002200  p=1.000000e+00 + i(2.220446e-16)
%11.402280  p=1.000000e+00 + i(-8.604228e-16)
%11.802360  p=1.000000e+00 + i(-3.608225e-16)
%12.202440  p=1.000000e+00 + i(-5.828671e-16)
%12.602521  p=1.000000e+00 + i(-1.554312e-15)
%13.002601  p=1.000000e+00 + i(5.551115e-17)
%13.402681  p=1.000000e+00 + i(4.163336e-16)
%13.802761  p=1.000000e+00 + i(6.661338e-16)
%14.202841  p=1.000000e+00 + i(-6.938894e-16)
%14.602921  p=1.000000e+00 + i(8.326673e-17)
15.003001  &p=1.000000$\times 10^0$ &+ i 8.049117$\times 10^{-16}$ \\
%15.403081  p=1.000000e+00 + i(1.415534e-15)
%15.803161  p=1.000000e+00 + i(7.494005e-16)
%16.203241  p=1.000000e+00 + i(0.000000e+00)
%16.603321  p=1.000000e+00 + i(1.498801e-15)
%17.003401  p=1.000000e+00 + i(-1.554312e-15)
%17.403481  p=1.000000e+00 + i(4.996004e-16)
%17.803561  p=1.000000e+00 + i(-3.608225e-16)
%18.203641  p=1.000000e+00 + i(2.498002e-16)
%18.603721  p=1.000000e+00 + i(1.526557e-15)
%19.003801  p=1.000000e+00 + i(-6.661338e-16)
%19.403881  p=1.000000e+00 + i(3.885781e-16)
%19.803961  p=1.000000e+00 + i(-7.494005e-16)
20.204041  &p=1.000000$\times 10^0$&+ i 1.137979$\times 10^{-15}$ \\
%20.604121  p=1.000000e+00 + i(-6.383782e-16)
%21.004201  p=1.000000e+00 + i(8.049117e-16)
%21.404281  p=1.000000e+00 + i(5.551115e-16)
%21.804361  p=1.000000e+00 + i(2.775558e-17)
%22.204441  p=1.000000e+00 + i(-3.053113e-16)
%22.604521  p=1.000000e+00 + i(2.775558e-17)
%23.004601  p=1.000000e+00 + i(-8.881784e-16)
%23.404681  p=1.000000e+00 + i(-9.714451e-16)
%23.804761  p=1.000000e+00 + i(-1.165734e-15)
%24.204841  p=1.000000e+00 + i(-1.165734e-15)
%24.604921  p=1.000000e+00 + i(5.551115e-16)
25.005001  &p=1.000000$\times 10^0$ &- i 2.164935$\times 10^{-15}$\\
\hline                                          
\end{tabular} 
\end{table} 

\vfill\eject

\begin{figure}
\caption{
Plots of the real and imaginary parts of the
half-shell transition
matrix computed using the Fourier transform of the path integral
calculation and by numerical solution of the Lippmann-Schwinger
equation ($p=5.0$).}
\begin{minipage}[t]{.45\linewidth}
\centering
%\includegraphics[scale=.5]{2steps_lc_re.pdf}
%\includegraphics[scale=.45]{fig_24.pdf}
\includegraphics[scale=.45]{fig_14a.pdf}
%\caption{Plots of the real part of the
%half-shell transition
%matrix computed using the Fourier transform of the path integral
%calculation and by numerical solution of the Lippmann-Schwinger
%equation ($k=5.$).}
%\caption{Re(T) exact vs path integral}
%\label{figure 20}
%\label{figure 24}
%\end{figure}  
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
%\begin{figure}
\centering
%\includegraphics[scale=.5]{2steps_lc_im.pdf}
%\includegraphics[scale=.45]{fig_25.pdf}
\includegraphics[scale=.45]{fig_14b.pdf}
%\caption{
%Plots of the imaginary part of the
%half-shell transition
%matrix computed using the Fourier transform of the path integral
%calculation and by numerical solution of the Lippmann-Schwinger
%equation ($k=5.$).}
%\caption{Im(T) exact vs. path integral}
%\label{figure 21}
%\label{figure 25}
\end{minipage}
\label{figure 14}
\end{figure}



\end{document}
$a=-\infty$
\beq
I[-\infty,b] = \int_{-\infty}^b e^{ix^2} dx =
{1 \over 2} \int_{-\infty}^{\infty} e^{ix^2} dx - \int_b^0 e^{ix^2} dx =
\sqrt{{\pi \over 2}} \left ({1+i \over 2} + C_c(b)+i S_c(b) \right ) .
\label{f.11}
\eeq
and for $b=\infty$
\beq
I[a,\infty] = \int_a^{\infty} e^{ix^2} dx =
{1 \over 2} \int_{-\infty}^{\infty} e^{ix^2} dx - \int_0^a e^{ix^2} dx =
\sqrt{{\pi \over 2}} \left ({1+i \over 2} - C_c(a)-i S_c(a) \right ) .
\label{f.11}
\eeq

\vfill\eject


\end{document}

\begin{figure}
\caption{Plots of the exact smeared Born approximation,
the real and imaginary parts of the Born approximation smeared
against the wave packet
computed using complex probabilities with V set to 0.} 
\centering
%\includegraphics[scale=.5]{fig_21.pdf}
%\includegraphics[scale=.5]{fig_21.pdf}
\includegraphics[scale=.5]{fig_12.pdf}
%\label{figure 21}
\label{figure 12}
%\label{figure 25}
\end{figure}  



\begin{figure}
\caption{Plots of the real and imaginary parts of the free wave packet
at time $t=0$ and the real and imaginary parts of the free path integral
evolved wave packet  
from $t=-3.0$ to $t=0.0$.}
\begin{minipage}[t]{.45\linewidth}
%\begin{subfigure}[t]{.45\linewidth}
\centering
%\includegraphics[scale=.5]{figure_2a.pdf}
%\includegraphics[scale=.5]{fig_4.pdf}
\includegraphics[scale=.5]{fig_2a.pdf}  
%%\caption{Re( $\langle x \vert \psi_{0i}(0) \rangle$)
%and Re($\langle x \vert U_0 (3) \vert \psi_{0i}(-3) \rangle$)
%using 30 time steps}
%\caption{real part}
%\label{figure 2}
%\end{figure}  
\end{minipage}
%\end{subfigure}
%\begin{subfigure}
\begin{minipage}[t]{.45\linewidth}
% \begin{figure}\
\centering
%\includegraphics[scale=.5]{figure_2b.pdf}  
%\includegraphics[scale=.5]{fig_5.pdf}
\includegraphics[scale=.5]{fig_2b.pdf}  
%\caption{Imaginary part}
%\caption{Im( $\langle x \vert \psi_{0i}(0) \rangle$)
%  and Im$\langle x \vert U_0(3) \vert \psi_{0i}(-3) \rangle$)
%using 30 time steps}
%\label{figure 3}
\end{minipage}
\label{figure 2}
%\end{subfigure}
\end{figure}  


\[
(U^N-1) \vert v_m\rangle =0
\]
\[
(U^N-1) = (U-1) \sum_{m=1}^N U^m = \sum_{m=1}^N (U^{m+1} - U^m)=0 
\]
since $U^{N+1}=U$.  We also have for $u_k= e^{{2\pi i k\over N}}$:
\[
(({U\over u_k})^N-1) = ({U\over u_k}-1) \sum_{m=1}^N ({U\over u_k})^m
\]
Completeness means
\[
\sum_{m=1}^N ({U\over u_k})^m = N \times \vert u_k \rangle \langle u_k \vert
\]
which gives
\[
\boxed{
\vert u_k \rangle \langle u_k \vert = {1 \over N}\sum_{m=1}^N {U\over u_k}^m
}
\]
Phase choices
\[
\langle a_N \vert u_k \rangle = \langle u_k \vert a_N \rangle := {1 \over \sqrt{N}}
\]
\[
\langle a_m \vert u_k \rangle \langle u_k \vert a_N \rangle =
\langle a_m \vert u_k \rangle {1 \over \sqrt{N}} =
{1 \over N} \sum_{n=1}^N \langle a_m \vert a_n \rangle {1 \over u_{kn}}=
{1 \over N} e^{-{2\pi i kn \over N}}
\]
\[
\boxed{
\langle a_m \vert u_k \rangle ={1 \over \sqrt{N}} e^{-{2\pi i kn \over N}}
}
\]




\section{Notation}
\[
A\vert v_n \rangle = a_n \vert v_n \rangle
\]
\[
U\vert v_n \rangle = \vert v_{n+1} \rangle
\qquad
U\vert v_N \rangle = \vert v_{1} \rangle
\]
\[
(U^N-1) \vert v_m\rangle =0
\]
\[
(U^N-1) = (U-1) \sum_{m=1}^N U^m = \sum_{m=1}^N (U^{m+1} - U^m)=0 
\]
since $U^{N+1}=U$.  We also have for $u_k= e^{{2\pi i k\over N}}$:
\[
(({U\over u_k})^N-1) = ({U\over u_k}-1) \sum_{m=1}^N ({U\over u_k})^m
\]
Completeness means
\[
\sum_{m=1}^N ({U\over u_k})^m = N \times \vert u_k \rangle \langle u_k \vert
\]
which gives
\[
\boxed{
\vert u_k \rangle \langle u_k \vert = {1 \over N}\sum_{m=1}^N {U\over u_k}^m
}
\]
Phase choices
\[
\langle a_N \vert u_k \rangle = \langle u_k \vert a_N \rangle := {1 \over \sqrt{N}}
\]
\[
\langle a_m \vert u_k \rangle \langle u_k \vert a_N \rangle =
\langle a_m \vert u_k \rangle {1 \over \sqrt{N}} =
{1 \over N} \sum_{n=1}^N \langle a_m \vert a_n \rangle {1 \over u_{kn}}=
{1 \over N} e^{-{2\pi i kn \over N}}
\]
\[
\boxed{
\langle a_m \vert u_k \rangle ={1 \over \sqrt{N}} e^{-{2\pi i kn \over N}}
}
\]
Next given $\vert u_k \rangle$ define
\[
\langle u_k \vert V = \langle u_{k+1} \vert 
\]
\[
\langle u_k \vert (V^N-1) = 0
\]
same argument gives
\[
\boxed{
\vert v_k \rangle \langle v_k \vert = {1 \over N}\sum_{m=1}^N {V\over v_k}^m
}
\]
Phase choice
\[
\langle u_N \vert v_k \rangle = \langle v_k \vert u_N \rangle := {1 \over \sqrt{N}}
\]
\[
\langle u_N \vert v_k \rangle \langle v_k \vert u_m \rangle =
\langle v_k \vert u_m \rangle {1 \over \sqrt{N}} =
{1 \over N} \sum_{n=1}^N \langle u_n \vert u_m \rangle {1 \over v_{kn}}=
{1 \over N} e^{-{2\pi i km \over N}}
\]
\[
\boxed{
\langle v_k \vert u_m \rangle ={1 \over \sqrt{N}} e^{-{2\pi i km \over N}}
}
\]
Comparing gives
\[
\vert a_n\rangle = \vert v_n \rangle
\]
\[
UV= U \sum_k \vert v_k\rangle e^{{2 \pi i k \over N}}\langle v_k \vert =
\sum_k \vert v_{k+1}\rangle e^{{2 \pi i (k+1) \over N}}\langle v_k \vert
e^{-{2 \pi i  \over N}}=
\]
\[
\sum_k \vert v_{k+1}\rangle e^{{2 \pi i (k+1) \over N}}\langle v_{k+1} \vert U
e^{-{2 \pi i  \over N}}= VU e^{-{2 \pi i  \over N}}
\]
or
\[
\boxed{
VU = UV e^{{2 \pi i \over N}}
}
\]
General operator expansions
\[
O = \sum_{mn} O_{mn} \vert v_m \rangle \langle v_n \vert =
\]
\[
\sum_{mn} O_{mn} U^{m-n} \vert v_n \rangle \langle v_n \vert =
\]
\[
\sum_{mnk} O_{mn} U^{m-n} {1 \over N} ({V\over v_n})^{k}=
\]
\[
\sum_{mnk} O_{mn} {e^{-{2\pi i nk \over N}}\over N} U^{m-n}V^k=
\]
let $l= m-n$,  $m= n+l$
\[
\sum_{lnk} O_{n+l,n} {e^{-{2\pi i nk \over N}}\over N} U^{l}V^k=
\]
\[
\sum_{lnk} O_{n+l,n} {e^{-{2\pi i nk \over N}}\over N} V^k U^l e^{-i{2 \pi ikl\over N}}
\]
\[
\sum_{lnk} O_{n+l,n} {e^{-{2\pi i (n+l)k \over N}}\over N} V^k U^l 
\]
let $m= n+l$ $n=m-l$
\[
\sum_{lmk} O_{m,m-l} {e^{-{2\pi i mk \over N}}\over N} V^k U^l 
\]

\[
\boxed{
O = \sum_{mn} O_{mn} \vert v_m \rangle \langle v_n \vert =
\sum_{lmk} O_{m,m-l} {e^{-{2\pi i mk \over N}}\over N} V^k U^l =
\sum_{lmk} O_{m+l,m} {e^{-{2\pi i mk \over N}}\over N} U^{l}V^k
}
\]
This can also be put in the $\vert u_k \rangle$ basis. Discrete momentum and
coordinate operators are defined by
\[
U= e^{i \epsilon q} \qquad  V= e^{i \epsilon p}
\]
where $\epsilon$ is a real parameter.
The eigenvalues of 
\[
U\vert u_n\rangle = e^{i 2\pi i n \over N} \vert u_n\rangle =
\vert u_n\rangle e^{i \epsilon q_n }
\]
\[
q_n = {2 \pi n \over \epsilon N}
\]
\end{document}
