Thermal quantum field theory

Last updated

In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature.

In the Matsubara formalism, the basic idea (due to Felix Bloch [1] ) is that the expectation values of operators in a canonical ensemble

may be written as expectation values in ordinary quantum field theory [2] where the configuration is evolved by an imaginary time . One can therefore switch to a spacetime with Euclidean signature, where the above trace (Tr) leads to the requirement that all bosonic and fermionic fields be periodic and antiperiodic, respectively, with respect to the Euclidean time direction with periodicity (we are assuming natural units ). This allows one to perform calculations with the same tools as in ordinary quantum field theory, such as functional integrals and Feynman diagrams, but with compact Euclidean time. Note that the definition of normal ordering has to be altered. [3] In momentum space, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies and, through the de Broglie relation, to a discretized thermal energy spectrum . This has been shown to be a useful tool in studying the behavior of quantum field theories at finite temperature. [4] [5] [6] [7] It has been generalized to theories with gauge invariance and was a central tool in the study of a conjectured deconfining phase transition of Yang–Mills theory. [8] [9] In this Euclidean field theory, real-time observables can be retrieved by analytic continuation. [10] The Feynman rules for gauge theories in the Euclidean time formalism, were derived by C. W. Bernard. [8]    

The Matsubara formalism, also referred to as imaginary time formalism, can be extended to systems with thermal variations. [11] [12] In this approach, the variation in the temperature is recast as a variation in the Euclidean metric. Analysis of the partition function leads to an equivalence between thermal variations and the curvature of the Euclidean space. [11] [12]

The alternative to the use of fictitious imaginary times is to use a real-time formalism which come in two forms. [13] A path-ordered approach to real-time formalisms includes the Schwinger–Keldysh formalism and more modern variants. [14] The latter involves replacing a straight time contour from (large negative) real initial time to by one that first runs to (large positive) real time and then suitably back to . [15] In fact all that is needed is one section running along the real time axis, as the route to the end point, , is less important. [16] The piecewise composition of the resulting complex time contour leads to a doubling of fields and more complicated Feynman rules, but obviates the need of analytic continuations of the imaginary-time formalism. The alternative approach to real-time formalisms is an operator based approach using Bogoliubov transformations, known as thermo field dynamics. [13] [17] As well as Feynman diagrams and perturbation theory, other techniques such as dispersion relations and the finite temperature analog of Cutkosky rules can also be used in the real time formulation. [18] [19]

An alternative approach which is of interest to mathematical physics is to work with KMS states.

See also

Related Research Articles

Hawking radiation is the theoretical thermal black-body radiation released outside a black hole's event horizon. This is counterintuitive because once ordinary electromagnetic radiation is inside the event horizon, it cannot escape. It is named after the physicist Stephen Hawking, who developed a theoretical argument for its existence in 1974. Hawking radiation is predicted to be extremely faint and is many orders of magnitude below the current best telescopes' detecting ability.

In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable.

<span class="mw-page-title-main">Partition function (quantum field theory)</span> Generating function for quantum correlation functions

In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral formalism. They are the imaginary time versions of statistical mechanics partition functions, giving rise to a close connection between these two areas of physics. Partition functions can rarely be solved for exactly, although free theories do admit such solutions. Instead, a perturbative approach is usually implemented, this being equivalent to summing over Feynman diagrams.

<span class="mw-page-title-main">Black hole thermodynamics</span> Area of study

In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black-body radiation led to the development of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.

The Unruh effect is a theoretical prediction in quantum field theory that an observer who is uniformly accelerating through empty space will perceive a thermal bath. This means that even in the absence of any external heat sources, an accelerating observer will detect particles and experience a temperature. In contrast, an inertial observer in the same region of spacetime would observe no temperature.

In physics, the Landau pole is the momentum scale at which the coupling constant of a quantum field theory becomes infinite. Such a possibility was pointed out by the physicist Lev Landau and his colleagues. The fact that couplings depend on the momentum scale is the central idea behind the renormalization group.

In theoretical physics, Euclidean quantum gravity is a version of quantum gravity. It seeks to use the Wick rotation to describe the force of gravity according to the principles of quantum mechanics.

<span class="mw-page-title-main">Hagen Kleinert</span> German physicist

Hagen Kleinert is professor of theoretical physics at the Free University of Berlin, Germany , Honorary Doctor at the West University of Timișoara, and at the Kyrgyz-Russian Slavic University in Bishkek. He is also Honorary Member of the Russian Academy of Creative Endeavors. For his contributions to particle and solid-state physics he was awarded the Max Born Prize 2008 with Medal. His contribution to the memorial volume celebrating the 100th birthday of Lev Davidovich Landau earned him the Majorana Prize 2008 with Medal. He is married to Dr. Annemarie Kleinert since 1974 with whom he has a son Michael Kleinert.

A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.

In gauge theory, especially in non-abelian gauge theories, global problems at gauge fixing are often encountered. Gauge fixing means choosing a representative from each gauge orbit, that is, choosing a section of a fiber bundle. The space of representatives is a submanifold and represents the gauge fixing condition. Ideally, every gauge orbit will intersect this submanifold once and only once. Unfortunately, this is often impossible globally for non-abelian gauge theories because of topological obstructions and the best that can be done is make this condition true locally. A gauge fixing submanifold may not intersect a gauge orbit at all or it may intersect it more than once. The difficulty arises because the gauge fixing condition is usually specified as a differential equation of some sort, e.g. that a divergence vanish. The solutions to this equation may end up specifying multiple sections, or perhaps none at all. This is called a Gribov ambiguity.

<span class="mw-page-title-main">Gary Gibbons</span> British theoretical physicist

Gary William Gibbons is a British theoretical physicist.

Malcolm John Perry is a British theoretical physicist and emeritus professor of theoretical physics at University of Cambridge and professor of theoretical physics at Queen Mary University of London. His research mainly concerns quantum gravity, black holes, general relativity, and supergravity.

In mathematical physics, a caloron is the finite temperature generalization of an instanton.

<span class="mw-page-title-main">Light front quantization</span> Technique in computational quantum field theory

The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is one Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and , are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others.

<span class="mw-page-title-main">Quantum triviality</span> Possible outcome of renormalization in physics

In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as quantum triviality. Strong evidence supports the idea that a field theory involving only a scalar Higgs boson is trivial in four spacetime dimensions, but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the Standard Model of particle physics, the question of triviality in Higgs models is of great importance.

Stuart Samuel is a theoretical physicist known for his work on the speed of gravity and for his work with Alan Kostelecký on spontaneous Lorentz violation in string theory, now called the Bumblebee model. He also made significant contributions in field theory and particle physics.

The hierarchical equations of motion (HEOM) technique derived by Yoshitaka Tanimura and Ryogo Kubo in 1989, is a non-perturbative approach developed to study the evolution of a density matrix of quantum dissipative systems. The method can treat system-bath interaction non-perturbatively as well as non-Markovian noise correlation times without the hindrance of the typical assumptions that conventional Redfield (master) equations suffer from such as the Born, Markovian and rotating-wave approximations. HEOM is applicable even at low temperatures where quantum effects are not negligible.

Periodic instantons are finite energy solutions of Euclidean-time field equations which communicate between two turning points in the barrier of a potential and are therefore also known as bounces. Vacuum instantons, normally simply called instantons, are the corresponding zero energy configurations in the limit of infinite Euclidean time. For completeness we add that ``sphalerons´´ are the field configurations at the very top of a potential barrier. Vacuum instantons carry a winding number, the other configurations do not. Periodic instantons were discovered with the explicit solution of Euclidean-time field equations for double-well potentials and the cosine potential with non-vanishing energy and are explicitly expressible in terms of Jacobian elliptic functions. Periodic instantons describe the oscillations between two endpoints of a potential barrier between two potential wells. The frequency of these oscillations or the tunneling between the two wells is related to the bifurcation or level splitting of the energies of states or wave functions related to the wells on either side of the barrier, i.e. . One can also interpret this energy change as the energy contribution to the well energy on either side originating from the integral describing the overlap of the wave functions on either side in the domain of the barrier.

In many-body physics, the problem of analytic continuation is that of numerically extracting the spectral density of a Green function given its values on the imaginary axis. It is a necessary post-processing step for calculating dynamical properties of physical systems from Quantum Monte Carlo simulations, which often compute Green function values only at imaginary times or Matsubara frequencies.

In quantum field theory, the Polyakov loop is the thermal analogue of the Wilson loop, acting as an order parameter for confinement in pure gauge theories at nonzero temperatures. In particular, it is a Wilson loop that winds around the compactified Euclidean temporal direction of a thermal quantum field theory. It indicates confinement because its vacuum expectation value must vanish in the confined phase due to its non-invariance under center gauge transformations. This also follows from the fact that the expectation value is related to the free energy of individual quarks, which diverges in this phase. Introduced by Alexander M. Polyakov in 1975, they can also be used to study the potential between pairs of quarks at nonzero temperatures.

References

  1. ↑ Bloch, F. (1932). "Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika". Z. Phys. 74 (5–6): 295–335. Bibcode:1932ZPhy...74..295B. doi:10.1007/BF01337791. S2CID   120549836.
  2. ↑ Jean Zinn-Justin (2002). Quantum Field Theory and Critical Phenomena. Oxford University Press. ISBN   978-0-19-850923-3.
  3. ↑ T.S. Evans and D.A. Steer (1996). "Wick's theorem at finite temperature". Nucl. Phys. B. 474 (2): 481–496. arXiv: hep-ph/9601268 . Bibcode:1996NuPhB.474..481E. doi:10.1016/0550-3213(96)00286-6. S2CID   119436816.
  4. ↑ D.A. Kirznits JETP Lett. 15 (1972) 529.
  5. ↑ D.A. Kirznits and A.D. Linde, Phys. Lett. B42 (1972) 471; it Ann. Phys. 101 (1976) 195.
  6. ↑ Weinberg, S. (1974). "Gauge and Global Symmetries at High Temperature". Phys. Rev. D. 9 (12): 3357–3378. Bibcode:1974PhRvD...9.3357W. doi:10.1103/PhysRevD.9.3357.
  7. ↑ L. Dolan, and R. Jackiw (1974). "Symmetry behavior at finite temperature". Phys. Rev. D. 9 (12): 3320–3341. Bibcode:1974PhRvD...9.3320D. doi:10.1103/PhysRevD.9.3320.
  8. 1 2 C. W. Bernard, Phys. Rev. D9 (1974) 3312.
  9. ↑ D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43.
  10. ↑ T.S. Evans (1992). "N-Point Finite Temperature Expectation Values at Real Times". Nucl. Phys. B. 374 (2): 340–370. arXiv: hep-ph/9601268 . Bibcode:1992NuPhB.374..340E. doi:10.1016/0550-3213(92)90357-H. S2CID   120072328.
  11. 1 2 S. Ganesh (2022). "Quantum theory, thermal gradients and the curved Euclidean space". International Journal of Modern Physics A. 37 (17). arXiv: 2206.13324 . Bibcode:2022IJMPA..3750125G. doi:10.1142/S0217751X22501251. S2CID   250073218.
  12. 1 2 Ganesh, S (2023-02-16). "5D thermal field theory, Einstein field equations and spontaneous symmetry breaking". Classical and Quantum Gravity. 40 (4): 045008. arXiv: 2301.04827v1 . doi:10.1088/1361-6382/acb24c. ISSN   0264-9381.
  13. 1 2 N.P. Landsman and Ch.G. van Weert (1987). "Real- and imaginary-time field theory at finite temperature and density". Physics Reports. 145 (3–4): 141–249. Bibcode:1987PhR...145..141L. doi:10.1016/0370-1573(87)90121-9.
  14. ↑ A.J. Niemi, G.W. Semenoff (1984). "Finite Temperature Quantum Field Theory in Minkowski Space". Annals of Physics. 152 (1): 105–129. Bibcode:1984AnPhy.152..105N. doi:10.1016/0003-4916(84)90082-4.
  15. ↑ Zinn-Justin, Jean (2000). "Quantum field theory at finite temperature: An introduction". arXiv: hep-ph/0005272 .
  16. ↑ T.S. Evans (1993). "New Time Contour for Equilibrium Real-Time Thermal Field-Theories". Phys. Rev. D. 47 (10): R4196–R4198. arXiv: hep-ph/9310339 . Bibcode:1993PhRvD..47.4196E. doi:10.1103/PhysRevD.47.R4196. PMID   10015491. S2CID   119486408.
  17. ↑ H. Chiu; H. Umezawa (1993). "A unified formalism of thermal quantum field theory". International Journal of Modern Physics A. 9 (14): 2363 ff. Bibcode:1994IJMPA...9.2363C. doi:10.1142/S0217751X94000960.
  18. ↑ R.L. Kobes, G.W. Semenoff (1985). "Discontinuities of Green Functions in Field Theory at Finite Temperature and Density". Nucl. Phys. B. 260 (3–4): 714–746. Bibcode:1985NuPhB.260..714K. doi:10.1016/0550-3213(85)90056-2.
  19. ↑ R.L. Kobes, G.W. Semenoff (1986). "Discontinuities of Green Functions in Field Theory at Finite Temperature and Density". Nucl. Phys. B. 272 (2): 329–364. Bibcode:1986NuPhB.272..329K. doi:10.1016/0550-3213(86)90006-4.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.