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  Does statistical mechanics equal one-loop quantum field theory?

Allen, B. (1986). Does statistical mechanics equal one-loop quantum field theory? Physical Review D, 33(12), 3640-3644. doi:10.1103/PhysRevD.33.3640.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-5DE5-2 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-5DE6-F
Genre: Journal Article

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 Creators:
Allen, Bruce1, Author              
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1Observational Relativity and Cosmology, AEI-Hannover, MPI for Gravitational Physics, Max Planck Society, ou_24011              

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 Abstract: The ‘‘thermodynamic’’ partition function ZT(β)=Jnexp(-βEn) is compared to the Euclidean ‘‘quantum’’ path integral ZQ(β)=Fd[φ]exp(-S) over (anti)periodic fields φ(τ+β)=±φ(τ). We assume (1) free spin-0 or spin-(1/2) fields and (2) an ultrastatic spacetime. Our main result is that ZT(β) does not equal ZQ(β). Nevertheless, they are simply related: we prove that lnZQ(β)=lnZT(β)+(A+B lnμ2)β. Thus, the logarithms of the two partition functions differ only by a term proportional to β. The constant A arises from vacuum energy and the constant B from the renormalization-scale (μ) dependence of ZQ. We derive a simple formula for A and B in terms of the ‘‘energy’’ ζ function ζE(z)=JkEk -z. In particular we show that A and B are determined by the behavior of the energy ζ function near z=-1: for small ε, ±ζE(-1+ε)=(1/4)Bε-1+( 1/2)A+O(ε) (where the upper sign applies to bosons and the lower sign applies to fermions). We also give a high-temperature expansion of Z(β) in terms of ζE(z). Finally we argue that ZT and ZQ are interchangeable in any situation where gravitational effects are unimportant. This is because adding a term linear in β to lnZ is equivalent to shifting all energies by a constant; but if gravity is neglected, then the physics only depends upon the difference between energies, which is unchanged.

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 Dates: 1986-02
 Publication Status: Published in print
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Title: Physical Review D
Source Genre: Journal
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Pages: - Volume / Issue: 33 (12) Sequence Number: - Start / End Page: 3640 - 3644 Identifier: ISSN: 1550-7998