Effective potential for non-convex potentials

Fujimoto, Yasushi; O’Raifeartaigh, L.; Parravicini, Giuseppe Pastori

doi:10.1016/0550-3213(83)90305-x

Cited by 134 publications

(94 citation statements)

References 14 publications

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“…A specific example of this behavior is spontaneous symmetry breaking in QFT, which causes the naive loop expansion to violate the convexity properties of the effective potential. This situation must be dealt with by an improved loop expansion [37][38][39].…”

Section: Stochastic Averages Characteristic Functional Feynmanmentioning

confidence: 99%

Effective action for stochastic partial differential equations

et al. 1999

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Stochastic partial differential equations (SPDEs) are the basic tool for modeling systems where noise is important. SPDEs are used for models of turbulence, pattern-formation, and the structural development of the universe itself. It is reasonably well-known that certain SPDEs can be manipulated to be equivalent to (non-quantum) field theories that nevertheless exhibit deep and important relationships with quantum field theory. In this paper we systematically extend these ideas: We set up a functional integral formalism and demonstrate how to extract all the one-loop physics for an arbitrary SPDE subject to arbitrary Gaussian noise. It is extremely important to realize that Gaussian noise does not imply that the field variables undergo Gaussian fluctuations, and that these non-quantum field theories are fully interacting. The limitation to one-loop is not as serious as might be supposed: Experience with quantum field theories (QFTs) has taught us that one-loop physics is often quite adequate to give a good description of the salient issues. The limitation to one loop does, however, offer marked technical advantages: Because at one loop almost any field theory can be rendered finite using zeta function technology, we can sidestep the complications inherent in the Martin-Siggia-Rose formalism (the SPDE analog of the BRST formalism used in QFT) and instead focus attention on a minimalist approach that uses only the physical fields (this "direct approach" is the SPDE analog of canonical quantization using physical fields.) After setting up the general formalism for the characteristic functional (partition function), we show how to define the effective action to all loops, and then focus on the one-loop effective action, and its specialization to constant fields: the effective potential. The physical interpretation of the effective action and effective potential for SPDEs is addressed and we show that key features carry over from QFT to the case of SPDEs. An important result is that the amplitude of the two-point function governing the noise acts as the loop-counting parameter and is the analog of Planck's constanth in this SPDE context. We derive a general expression for the one-loop effective potential of an arbitrary SPDE subject to translation-invariant Gaussian noise, and compare this with the one-loop potential for QFT.

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Section: Stochastic Averages Characteristic Functional Feynmanmentioning

confidence: 99%

Effective action for stochastic partial differential equations

et al. 1999

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“…We must also keep in mind the fact that the bare theory (i.e., defined by the action (12)) does not depend on the arbitrary scale µ introduced by the renormalization scheme. Therefore, just as for the case of QFTs [18,19], we will derive a set of equations that govern the scale dependence of the parameters appearing in the RD effective action from the identity…”

Section: One-loop Renormalizationmentioning

confidence: 99%

“…In the quantum domain, one is interested in computing quantities such as the effective action and the effective potential , which provide crucial information regarding the structure of the underlying theory at different length and time scales and are important in assessing the theory's renormalizability (or lack thereof), the determination of the running of couplings and parameters, patterns of spontaneous and dynamical symmetry breaking, and the structure of short distance (ultraviolet) and long distance (infrared) divergences [13][14][15][16][17]. Moreover, for renormalizable theories, the computation of the effective action (actually, only its divergent part is needed) can be used to extract the RGEs that govern the scale dependence of the couplings and parameters appearing in the theory [18][19][20]. Though perhaps better known in the context of these fields, these same techniques can be generalized and applied to reveal the corresponding one-loop physics associated with stochastic dynamic phenomena and to systems subject to fluctuations.…”

Section: Introductionmentioning

confidence: 99%

Heat kernel regularization of the effective action for stochastic reaction-diffusion equations

Hochberg

Molina-Parı́s²,

Visser³

2001

Phys. Rev. E

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The presence of fluctuations and non-linear interactions can lead to scale dependence in the parameters appearing in stochastic differential equations. Stochastic dynamics can be formulated in terms of functional integrals. In this paper we apply the heat kernel method to study the short distance renormalizability of a stochastic (polynomial) reaction-diffusion equation with real additive noise. We calculate the one-loop effective action and its ultraviolet scale dependent divergences. We show that for white noise a polynomial reaction-diffusion equation is one-loop finite in d = 0 and d = 1, and is one-loop renormalizable in d = 2 and d = 3 space dimensions. We obtain the one-loop renormalization group equations and find they run with scale only in d = 2. PACS number(s): 02.50.Ey;02.50.-r;05.40.+j

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“…A better approximation is obtained by a superposition of two Gaussians, centered at Φ = ±v and weighted so that Φ = φ. This procedure corresponds, effectively, to a Maxwell construction of the true effective potential [3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Quantum fluctuations and dynamical chaos: An effective potential approach

Matinyan

Müller

1997

Found Phys

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We discuss the intimate connection between the chaotic dynamics of a classical field theory and the instability of the one-loop effective action of the associated quantum field theory. Using the example of massless scalar electrodynamics, we show how the radiatively induced spontaneous symmetry breaking stabilizes the vacuum state against chaos, and we speculate that monopole condensation can have the same effect in non-Abelian gauge theories.

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Effective potential for non-convex potentials

Cited by 134 publications

References 14 publications

Effective action for stochastic partial differential equations

Effective action for stochastic partial differential equations

Heat kernel regularization of the effective action for stochastic reaction-diffusion equations

Quantum fluctuations and dynamical chaos: An effective potential approach

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