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More precisely, the determinant can be shown to have the form c42 [Formula Presented] and thus contributes an additional term [Formula Presented] to the action [Formula Presented] However, as discussed in c66c67 c68 c69 the term [Formula Presented] precisely cancels acausal contributions in a perturbation expansion in λ, i.e., closed response loops. This also follows from a direct expansion of the Langevin equation (4.1) in powers of λ combined with successive noise contractions. Here response loops are absent due to the causal character of the unperturbed propagator [Formula Presented] Thus with this proviso we are entitled to use the identity (4.8)
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More precisely, the determinant can be shown to have the form 42 J=exp-(1/2)Trxt[ν∇2+λ(u∇u+∇u)] and thus contributes an additional term GJ to the action G. However, as discussed in 66676869 the term GJ=ln J precisely cancels acausal contributions in a perturbation expansion in λ, i.e., closed response loops. This also follows from a direct expansion of the Langevin equation (4.1) in powers of λ combined with successive noise contractions. Here response loops are absent due to the causal character of the unperturbed propagator (∂/∂t-ν∇2)-1. Thus with this proviso we are entitled to use the identity (4.8).
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101
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85036245944
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The connection between the Fokker-Planck description of the noisy Burgers equation c9, c56, entailing the stationary distribution [Formula Presented] and the present path integral formulation is clearly of interest and will be pursued in another context; otherwise, we refer to c42 for a general discussion
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The connection between the Fokker-Planck description of the noisy Burgers equation 956, entailing the stationary distribution P(u), and the present path integral formulation is clearly of interest and will be pursued in another context; otherwise, we refer to 42 for a general discussion.
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102
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85036202378
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The choice of Poisson algebra is to some extent arbitrary. The present choice has been dictated by the analogy between the present “stochastic” path integral and the usual Feynman phase space integral
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The choice of Poisson algebra is to some extent arbitrary. The present choice has been dictated by the analogy between the present “stochastic” path integral and the usual Feynman phase space integral.
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108
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85036297337
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writing down the “quantum Hamiltonian” [Formula Presented] we have chosen a normal ordering of the operators [Formula Presented] [Formula Presented] in the interaction term, i.e., we have placed the “momentum” [Formula Presented] to the left of the “coordinate” [Formula Presented]. In the present heuristic discussion of the “quantum mechanics” the ordering is, however, immaterial; otherwise see c42
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In writing down the “quantum Hamiltonian” Ĥ we have chosen a normal ordering of the operators û and φ^ in the interaction term, i.e., we have placed the “momentum” û to the left of the “coordinate” φ^. In the present heuristic discussion of the “quantum mechanics” the ordering is, however, immaterial; otherwise see 42.
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109
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85036323639
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writing down the Heisenberg equations of motion we have in the usual way gone from the Schrödinger picture with tine evolution operator [Formula Presented] to the Heisenberg picture with operator time evolution [Formula Presented] note that this transformation does not require [Formula Presented] to be Hermitian or [Formula Presented] to be unitary
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In writing down the Heisenberg equations of motion we have in the usual way gone from the Schrödinger picture with tine evolution operator Û(t)=exp[-i(Δ/ν)Ĥt] to the Heisenberg picture with operator time evolution û(t)=exp[+i(Δ/ν)Ĥt]û(0)exp[-i(Δ/ν)Ĥt]; note that this transformation does not require Ĥ to be Hermitian or Û(t) to be unitary.
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110
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We note that the Heisenberg equations of motion “drive the operators away from hermiticity” since [Formula Presented] is non-Hermitian
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We note that the Heisenberg equations of motion “drive the operators away from hermiticity” since Ĥ is non-Hermitian.
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