Derivation of the Lee-Yang term via stochastic quantization

Jose A. Magpantay; Danilo M. Yanga

doi:10.1103/PhysRevD.32.516

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Derivation of the Lee-Yang term via stochastic quantization

Jose A. Magpantay and Danilo M. Yanga

National Institute of Physics, College of Science, University of the Philippines, Diliman, Quezon City 3004, Philippines

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Phys. Rev. D 32, 516 – Published 15 July, 1985

DOI: https://doi.org/10.1103/PhysRevD.32.516

Abstract

We derive the so called Lee-Yang term for velocity-dependent potentials via stochastic quantization. The derivation shows that there is a subtlety in choosing the appropriate Langevin equation, i.e., the Langevin equation must have a positive-definite Fokker-Planck Hamiltonian. In the derivation, we also make use of the Stratonovic calculus to transform a stochastic equation with an additive white noise to a stochastic equation with multiplicative white noise.

References (7)

T. D. Lee and C. N. Yang, Phys. Rev. 128, 2082 (1962).
See, for example, E. S. Abers and B. W. Lee, Phys. Rep. 9C, 1 (1973).
G. Parisi and Wu Yong-Shi, Sci. Sin. 24, 483 (1981).
We assume here that the transformation from q to x is a transformation from curvilinear to Cartesian coordinates.
L. Stratonovic, J. Control Optim. 4, 362 (1966); R. E. Mortensen, J. Stat. Phys. 1, 271 (1969); K. L. C. Hunt and J. Ross, J. Chem. Phys. 75, 976 (1981).
Had we used the Ito calculus [K. Ito, Mem. Am. Math Soc. 4, 1 (1951); see also Hunt and Ross in Ref. 5], we would get instead the Langevin equation up 20 { partial $q_{i}$ } over { partial τ } + $h_{ik}$ $h_{jk}$ up 20 { δ $S_{E}$ } over { δ $q_{j}$ } - up 20 { partial $h_{kj}$ } over { partial $q_{j}$ } $h_{ik}$ = $h_{ik}$ $η_{k}$ ~~. However, as argued by Hunt et al.(see Ref. 4) the choice of the calculus is equivalent to choosing the parameter alpha in deriving the path-integral representation for the distribution function. This parameter alpha eventually drops out in the final expression for the distribution function; i.e., the distribution function is eventually independent of the calculus used.
M. Namiki, I. Ohba and K. Okano, Prog. Theor. Phys. 72, 350 (1984).

[c1] T. D. Lee and C. N. Yang, Phys. Rev. 128, 2082 (1962).

[c2] See, for example, E. S. Abers and B. W. Lee, Phys. Rep. 9C, 1 (1973).

[c3] G. Parisi and Wu Yong-Shi, Sci. Sin. 24, 483 (1981).

[c4] We assume here that the transformation from q to x is a transformation from curvilinear to Cartesian coordinates.

[c5] L. Stratonovic, J. Control Optim. 4, 362 (1966); R. E. Mortensen, J. Stat. Phys. 1, 271 (1969); K. L. C. Hunt and J. Ross, J. Chem. Phys. 75, 976 (1981).

[c6] Had we used the Ito calculus [K. Ito, Mem. Am. Math Soc. 4, 1 (1951); see also Hunt and Ross in Ref. 5], we would get instead the Langevin equation up 20 { partial $q_{i}$ } over { partial τ } + $h_{ik}$ $h_{jk}$ up 20 { δ $S_{E}$ } over { δ $q_{j}$ } - up 20 { partial $h_{kj}$ } over { partial $q_{j}$ } $h_{ik}$ = $h_{ik}$ $η_{k}$ ~~. However, as argued by Hunt et al.(see Ref. 4) the choice of the calculus is equivalent to choosing the parameter alpha in deriving the path-integral representation for the distribution function. This parameter alpha eventually drops out in the final expression for the distribution function; i.e., the distribution function is eventually independent of the calculus used.

[c7] M. Namiki, I. Ohba and K. Okano, Prog. Theor. Phys. 72, 350 (1984).

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Derivation of the Lee-Yang term via stochastic quantization

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