Euclidean formulation of relativistic quantum mechanics

W. N. Polyzou^{1} and Philip Kopp^{1}

1Department of Physics and Astronomy, Iowa City, IA 52246

We discuss preliminary work on a formulation of relativistic quantum mechanics that uses reflection-positive Euclidean Green functions or generating functionals as phenomenological input. This work is motivated by the Euclidean axioms of quantum field theory [?][?]. The key observations are (1) locality is not used to reconstruct the quantum theory and (2) it is possible to construct a fully relativistic quantum theory without performing an explicit analytic continuation.

Hilbert space vectors are represented by wave functionals A[ϕ] with inner product

A[ϕ] = ∑_{j=1}^{n_{a}} a_{j}e^{iϕ(f_{j})} ⟨A|B⟩ := ∑_{j,k}^{n_{a},n_{b}} a_{l}^{∗}b_{k}Z[g_{g} − Θf_{j}]

where a_{j} are complex constants, f_{j} are real Schwartz functions on 4 dimensional Euclidean space with positive-time support, Θ is the Euclidean time-reflection operator, and Z[f] is the Euclidean generating functional. Reflection positivity is the condition that ⟨A|A⟩ ≥ 0. For β ≥ 0 and a ∈ ℝ^{3} we define

T (β, a)A[ϕ] := ∑_{j=1}^{n_{a}} a_{j}e^{iϕ(f_{j,β,a})} f_{j,β,a}(τ, x) := f_{j}(τ − β, x − a).

The square of the mass operator operating on a wave functional A[ϕ] is

M^{2}A[ϕ] = (\frac{∂^{2}}{∂β^{2}} + \frac{∂^{2}}{∂a^{2}} ) T (β, a)A[ϕ]_{|β=a=0}.

Solutions of the mass eigenvalue problem with eigenvalue λ can be expanded in terms of an orthonormal set of wave functionals A_{n}[ϕ], ⟨A_{n}|A_{m}⟩ = δ_{mn}:

Ψ_{λ}[ϕ] = ∑ α_{n}A_{m}[ϕ].

Simultaneous eigenstates of mass, linear momentum, spin, and z component of spin can be constructed from Ψ_{λ}[ϕ] using

Ψ_{λ,j,p,μ}[ϕ] = ∫_{SU(2)} dR ∫_{ℝ^{3}} \frac{da}{(2π)^{3/2}} e^{−ip·Ra}U(R)T (0, a)Ψ_{λ}[ϕ]D_{μj}^{j∗}[R]

where U(R) rotates the vector arguments of f_{j}(τ, x) in A[ϕ]. When λ is in the discrete spectrum of M, Ψ_{λ,j,p,μ}[ϕ] is a wave functional for a single-particle state that necessarily transforms as a mass λ spin j irreducible representation.

Products of suitably normalized single-particle wave functionals define mappings from the product of single-particle irreducible representation spaces of the Poincaré group to the model Hilbert space. Because these wave functionals create only single particle states out of the vacuum, their products are Haag-Ruelle injection operators [?][?] for the two-Hilbert-space formulation [?] of scattering theory. If we define Φ[ϕ] := ∏_{k} Ψ_{λ_{k},j_{k},p_{k},μ_{k}}[ϕ], ⊗g_{k} = ∏ g_{k}(p_{k}, μ_{k}), and H_{f} = ∑_{k} \sqrt{λ_{k}^{2} + p_{k}^{2}}, then scattering wave operator can be defined by the limit

Ω_{±}| ⊗ g_{k}⟩ := lim_{t→±∞} e^{iHt}Φe^{−iH_{f} t}| ⊗ g_{k}⟩.

Using the Kato-Birman invariance principle [?] to replace H by −e^{−βH} gives

Ω_{±}| ⊗ g_{k}⟩ := lim_{n→±∞} e^{−ine^{−βH}}Φe^{ine^{−βH_{f}}} | ⊗ g_{k}⟩.

Since the spectrum of e^{−βH} is compact, for large fixed n e^{−ine^{−βH}} can be uniformly approximated by a polynomial in e^{−βH}, which is easy to calculate in this framework. These steps provide a means to construct all single-particle states, all scattering states, and compute the action of the Poincaré group on all single-particle states and S-matrix elements, using only the Euclidean generating functional as input. The advantages of this framework are the relative ease with which cluster properties can be satisfied, the close relation to the quantum mechanical interpretation of quantum field theory, and the ability to perform calculations directly in Euclidean space without analytic continuation.

We tested the general method for calculating scattering observables using a solvable quantum mechanical model of the two-nucleon system. These test calculations, which used narrow wave packets, a large finite n and a Chebyshev polynomial expansion of e^{inx}, exhibited convergence to the exact transition matrix elements for a range of relative momenta between about 100 MeV up to 2 GeV. This success warrants further investigation of this framework.

This research was supported by the U.S. Department of Energy.

References

[1] K. Osterwalder and R. Schrader, Comm. in Math. Phys. 31,83(1975).

[2] J. Fröhlich, Helvetica Physica Acta, 47,265(1974).

[3] R. Jost, The General Theory of Quantized Fields, AMS, 1965.

[4] M. Reed and B. Simon, Functional Analysis, Vol III, Scattering Theory, Academic Press, 1979.