SPACETIME COARSE GRAININGS IN RELATIVISTIC PARTICLE MOTION

JOHN T. WHELAN

Department of Physics, University of California, Santa Barbara, California 93106-9530, USA

ABSTRACT Hartle’s generalized quantum mechanics formalism is used to examine a simple spacetime coarse graining, i.e., a set of alternatives defined with respect to regions extended in time as well as space, in the quantum mechanics of a free relativistic particle. For suitable initial conditions, tractable formulas are found for branch wave functions. Some initial conditions are found to give decoherence and allow the consistent assignment of probabilities.

1. Introduction

Most formulations of quantum mechanics make predictions for alternatives defined “at a moment of time”, or more generally on a spacelike surface. However, in a quantum theory of gravity, in which the metric itself is behaving quantummechanically, the notion of “spacelike” should be ill-defined. It is constructive, then, to consider a theory which predicts spacetime alternatives, defined with respect to regions extended in time as well as space. Spacetime alternatives in nonrelativistic quantum mechanics have been considered in the past by Feynman,^{1} Yamada and Takagi,^{2} and Hartle.^{3} (And also in this volume by Yamada.) The present work examines spacetime alternatives for the quantum mechanics of a free relativistic particle in Minkowski space, which should be a closer analogy to gravity because it exhibits a single reparametrization invariance, which can be thought of as a subset of the diffeomorphism invariance exhibited by general relativity.

We use Hartle’s generalized quantum mechanics formalism^{4} in which the possible histories of the system (“fine-grained histories”) are partitioned into classes {c_{α}} (“coarse graining”). For each coarse graining, a complex matrix D(α, α′) (“decoherence functional”) is constructed according to certain conditions. If its off-diagonal elements vanish to some accuracy [D(α, α′) ≈ δ_{αα′}p(α)] [“(medium) decoherence”], quantum-mechanical interference is negligible and the diagonal elements are the probabilities of the alternatives to the same accuracy. We construct the decoherence functional by a sum over histories; the fine-grained histories over which we sum are arbitrary paths through spacetime, not just those which are single-valued in the “time” coördinate x^{0}. This construction is Lorentz invariant.

Here we present the decoherence functional for one simple coarse-graining which decoheres for certain initial and final conditions. For more details and a wider consideration of the subject, the reader is directed to the author’s recent paper.^{5}

2. Formulation^{6}

For each class c_{α} of paths, a restricted propagator K_{α} is constructed via a sum over those histories which start at a spacetime point x′ on an initial spacelike 3surface σ′, end at a point x′′ on a final surface σ′′ and are in the class c_{α}:

K_{α}(x′′, x′) = \frac{1}{2mi} ∫_{0}^{∞} dN ∫_{x′αx′′} Dx Dp exp [i ∫_{0}^{N} dτ (p · \frac{dx}{dτ} − \frac{p^{2} + m^{2}}{2m} )] . (1)

The sum of all the {K_{α}} in a coarse graining gives the Feynman propagator: ∑_{α} K_{α}(x′′, x′) = Δ_{F} (x′′ − x′). The initial condition is described by a Klein-Gordon wave function ψ(x′); to each alternative c_{α} there corresponds a branch wave function ψ_{α}(x′′) = iK_{α}(x′′, x′) ∘ ψ(x′) where ∘ is the Klein-Gordon inner product on σ′. Summing all the {ψ_{α}} gives the positive frequency component of ψ: ∑_{α} ψ_{α}(x′′) = iΔ_{F} (x′′−x′)∘ψ(x′) = ψ^{+}(x′′). The final condition is a mixed state described by a set of Klein-Gordon wave functions {φ_{i}(x′′)} and non-negative weights {p′′_{i}}, attached with the Klein-Gordon inner product on σ′′ to produce the decoherence functional:

D(α, α′) = \frac{∑_{i}(ψ_{α′} ∘ φ_{i})p′′_{i}(φ_{i} ∘ ψ_{α})}{∑_{i}(ψ^{+} ∘ φ_{i})p′′_{i}(φ_{i} ∘ ψ^{+})} = \frac{ψ_{α′} ∘ ρ′′ ∘ ψ_{α}}{ψ^{+} ∘ ρ′′ ∘ ψ^{+}} .  (2)

We would like to choose for ρ′′ a condition of future indifference, i.e., a completely unspecified final condition. This condition is usually implemented by replacing the final density matrix with the identity operator. However, for the (nonpositive-definite) Klein-Gordon inner product, the identity operator is given by

I(x_{2}, x_{1}) ≅ ∫ \frac{d^{3}p}{(2π)^{3}2ω_{p}} e^{ip·(x_{2}−x_{1})} (e^{−iω_{p}(t_{2}−t_{1})} − e^{iω_{p}(t_{2}−t_{1})}) ,  (3)

which is unacceptable as a final density matrix, since some of the weights {p′′_{i}} it implies are negative. Instead, we must take the final condition to be

ρ′′(x_{2}, x_{1}) = ∫ \frac{d^{3}p}{(2π)^{3}2ω_{p}} e^{ip·(x_{2}−x_{1})} (e^{−iω_{p}(t_{2}−t_{1})} + e^{iω_{p}(t_{2}−t_{1})}) ,  (4)

so that ψ_{α′} ∘ ρ′′ ∘ ψ_{α} = ψ_{α′}^{+} ∘ ψ_{α}^{+} − ψ_{α′}^{−} ∘ ψ_{α}^{−} = ψ_{α′} ∘ ψ_{α} − 2ψ_{α′}^{−} ∘ ψ_{α}^{−}, which differs from ψ_{α′} ∘ ψ_{α} whenever ψ_{α} and ψ_{α′} have negative frequency components.

3. Coarse Graining and Results

Let n be a constant spacelike unit vector (n·n = 1), and x_{n} = n·x be the component of x along n. We can use the timelike 3-surface x_{n} = 0 to define a set of three alternatives: c_{ℓ}, in which the entire path lies in the region x_{n} < 0; c_{r}, in which the path is confined to the region x_{n} > 0; and c_{b}, in which the path crosses x_{n} = 0. If we define the reflection of x through the plane x_{n} = 0 by x_{c} = x−2x_{n}n, it is possible to show by the method of images^{5} that K_{ℓ}(x′′, x′) = Θ(−x′′_{n}) [Δ_{F} (x′′−x′) − Δ_{F} (x′′−x′_{c})] Θ(−x′_{n}) and K_{r}(x′′, x′) = Θ(x′′_{n}) [Δ_{F} (x′′−x′) − Δ_{F} (x′′−x′_{c})] Θ(x′_{n}); the third restricted propagator can be found by superposition: K_{b} = Δ_{F} − K_{ℓ} − K_{r}. If we choose the initial state ψ(x′) to be antisymmetric about x_{n} = 0, (i.e., ψ(x_{c}) = −ψ(x)), the branch wave functions take the particularly simple forms ψ_{ℓ}(x′′) = Θ(−x′′_{n})ψ^{+}(x′′), ψ_{r}(x′′) = Θ(x′′_{n})ψ^{+}(x′′), and ψ_{b}(x′′) = 0. If the initial state is normalized so that ψ^{+} ∘ ψ^{+} = 1, the decoherence functional is then

( \begin{matrix}D(ℓ, ℓ) = \frac{1}{2} + ΔD & D(ℓ, r) = −ΔD & D(ℓ, b) = 0 \\ D(r, ℓ) = −ΔD & D(r, r) = \frac{1}{2} + ΔD & D(r, b) = 0 \\ D(b, ℓ) = 0 & D(b, r) = 0 & D(b, b) = 0 \end{matrix} )  (5)

where ΔD = −2ψ_{ℓ}^{−} ∘ ψ_{ℓ}^{−} = −2ψ_{r}^{−} ∘ ψ_{r}^{−} = 2ψ_{ℓ}^{−} ∘ ψ_{r}^{−} = 2ψ_{r}^{−} ∘ ψ_{ℓ}^{−} is given by^{a}
ΔD = 2 ∫ \frac{dk_{1n}dk_{2n}d^{2}k_{⊥}}{(2π)^{2}} \frac{ω_{1} + ω_{2}}{2\sqrt{ω_{1}ω_{2}}} ψ̃^{+}(k_{2})^{∗}ψ̃^{+}(k_{1})\frac{e^{−i(ω_{1}−ω_{2})t′′}}{k_{1n} − k_{2n}} ln (\frac{ω_{1} − k_{1n}}{ω_{2} − k_{2n}} ) . (6)

Note that whenever the alternatives do decohere (ΔD ≈ 0), the probabilities are given by p(ℓ) ≈ 1/2 ≈ p(r), p(b) = 0.

One special initial condition is for ψ̃^{+} to be peaked with a small width δk around a single wavenumber k (and its reflection k_{c}). Then ΔD becomes

ΔD = −4\frac{δk}{(2π)^{3/2}} [\frac{1}{ω_{0}} − \frac{1}{k_{0n}} ln (\frac{ω_{0} − k_{0n}}{ω_{0⊥}} )] + O([δk]^{2}),  (7)

and we have approximate decoherence to lowest order in δk.

Acknowledgements

The author wishes to thank K. V. Kuchař, N. Yamada, R. S. Tate, and D. A. Craig; the Isaac Newton Institue in Cambridge, England; and especially J. B. Hartle for advice, direction, and encouragement. This material is based upon work supported under a National Science Foundation Graduate Research Fellowship. This work was also supported by NSF grant PHY90-08502.

References

1. R. P. Feynman, Rev. Mod. Phys. 20 (1948) 367.

2. N. Yamada and S. Takagi, Prog. Theor. Phys. 87 (1992) 77. See also N. Yamada and S. Takagi, Prog. Theor. Phys. 85 (1991) 985; 86 (1991) 599.

3. J. B. Hartle, Phys. Rev. D44 (1991) 3173.

4. See, for example, J. B. Hartle, in Quantum Cosmology and Baby Universes, Proceedings of the 7th Jerusalem Winter School, 1989, edited by S. Coleman, J. Hartle, T. Piran, and S. Weinberg (World Scientific, Singapore, 1991).

5. J. T. Whelan, “Spacetime alternatives in the quantum mechanics of a relativistic particle”, to appear in Phys. Rev. D (1994); preprint gr-qc/9406029.

6. For more details, see J. B. Hartle, in Gravitation and Quantizations, Proceedings of the 1992 Les Houches Summer School, edited by B. Julia and J. Zinn-Justin (North Holland, Amsterdam, 1994).

aWe use here several pieces of notation, namely v_{⊥} = v − v_{n}n and ω_{⊥} = \sqrt{k_{⊥}^{2} + m^{2}}, and also that ψ̃^{+} is the Fourier transform of the positive energy part of ψ. We are also working in a reference frame where n has no time component and with a final surface σ′′ which is a surface of constant time t′′.