Abstract
Scalar fields describe spinless particles, and from an algebraic point of view they are relatively simple. In the electroweak theory a doublet of scalar fields form the Higgs sector and interact with the fields of leptons, quarks and gauge bosons. Scalar fields play a pivotal role in many inflationary cosmological scenarios—they could be responsible for the anticipated exponential expansion of the early universe. This chapter is devoted to the quantization of (Euclidean) scalar field theory at zero and finite temperature. With heat kernel and zeta-function methods we calculate the thermodynamic potential of non-interacting spinless particles. In the second part we introduce and discuss the Schwinger function and effective potential to investigate the phases of interacting scalar field theory at zero and finite temperature. This includes an exhaustive discussion of the Legendre–Frenchel transformation. At the end we discretize the scalar field theory on a space-time lattice and calculate the propagator for non-interacting scalars on a hyper-cubic lattice.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This should not be confused with the n-point Schwinger functions at zero temperature.
- 2.
This means that its square is the identity, i.e. if a the Legendre transform takes f to g, then the Legendre transform of g will be f.
References
M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, 1995)
L. Brown, Quantum Field Theory (Cambridge University Press, Cambridge, 1992)
R. Haag, Local Quantum Physics: Fields, Particles and Algebras (Springer, Berlin, 1996)
M. Maggiore, A Modern Introduction to Quantum Field Theory (Oxford University Press, Oxford, 2005)
J. Fröhlich, On the triviality of \(\lambda\phi_{d}^{4}\) theories and the approach to the critical point in d≥4 dimensions. Nucl. Phys. B 200, 281 (1982)
R. Fernandez, J. Fröhlich, A.D. Sokal, Random Walks, Critical Phenomena and Triviality in Quantum Field Theory (Springer, Berlin, 1992)
A. Liddle, D. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, Cambridge, 2000)
V. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, Cambridge, 2005)
K. Osterwalder, R. Schrader, Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83 (1973)
K. Osterwalder, R. Schrader, Axioms for Euclidean Green’s functions II. Commun. Math. Phys. 42, 281 (1975)
J. Fröhlich, Schwinger functions and their generating functionals. Helv. Phys. Acta 47, 265 (1974)
J. Glimm, A. Jaffe, Quantum Physics—A Functional Integral Point of View (Springer, New York, 1981)
J.S. Dowkder, R. Critchley, Effective Lagrangian and energy–momentum tensor in de Sitter space. Phys. Rev. D 13, 3224 (1976)
S.W. Hawking, Zeta-function regularization of path integrals in curved space. Commun. Math. Phys. 55, 133 (1977)
E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994)
H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers. Rend. Circ. Mat. Palermo 39, 1 (1915)
S. Blau, M. Visser, A. Wipf, Determinants, Dirac operators and one-loops physics. Int. J. Mod. Phys. A 4, 1467 (1989)
S. Weinberg, Gauge and global symmetries at high temperature. Phys. Rev. D 9, 3357 (1974)
L. Dolan, R. Jackiw, Symmetry behavior at finite temperature. Phys. Rev. D 9, 3320 (1974)
C.W. Bernard, Feynman rules for gauge theories at finite temperature. Phys. Rev. D 9, 3312 (1974)
J.I. Kapusta, Finite-Temperature Field Theory (Cambridge University Press, Cambridge, 2006)
M. Le Bellac, Thermal Field Theory (Cambridge University Press, Cambridge, 2000)
P.B. Gilkey, Invariance Theory: The Heat Equation and the Atiyah–Singer Index Theorem (CRC Press, Boca Raton, 1994)
D.V. Vassilevich, Heat kernel expansion: user’s manual. Phys. Rep. 388, 279 (2003)
S. Blau, M. Visser, A. Wipf, Zeta functions and the Casimir energy. Nucl. Phys. B 310, 163 (1988)
E. Elizalde, A. Romeo, Expressions for the zeta function regularized Casimir energy. J. Math. Phys. 30, 1133 (1989)
I. Sachs, A. Wipf, Finite temperature Schwinger model. Helv. Phys. Acta 65, 652 (1992)
E. Corrigan, P. Goddard, H. Osborn, S. Templeton, Zeta function regularization and multi—instanton determinants. Nucl. Phys. B 159, 469 (1979)
H.W. Braden, Expansion for field theories on S 1×Σ. Phys. Rev. D 25, 1028 (1982)
L. O’Raifeartaigh, A. Wipf, H. Yoneyama, The constraint effective potential. Nucl. Phys. B 271, 653 (1986)
H. Casimir, On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet., Ser. B Phys. Sci. 51, 793 (1948)
S.K. Lamoreaux, Demonstration of the Casimir force in the 0.6 to 6 μm range. Phys. Rev. Lett. 78, 5 (1997)
G. Bressi, G. Carugno, R. Onofrio, G. Ruoso, Measurement of the Casimir force between parallel metallic surfaces. Phys. Rev. Lett. 88, 041804 (2002)
M. Bordag, U. Mohideen, V.M. Mostepanenko, New developments in the Casimir effect. Phys. Rep. 353, 1 (2001)
K.A. Milton, The Casimir Effect (World Scientific, Singapore, 2001)
J.L. Cardy, I. Peschel, Finite size dependence of the free energy in two-dimensional critical systems. Nucl. Phys. B 300, 377 (1988)
C. Wiesendanger, A. Wipf, Running coupling constants from finite size effects. Ann. Phys. 233, 125 (1994)
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, London, 2002)
R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)
R. Balian, From Microphysics to Macrophysics, vol. 1 (Springer, Berlin, 1991)
J.B. Hiriat-Urruty, J.E. Matinez-Legaz, New formulas for the Legendre–Frenchel transform. J. Math. Anal. Appl. 288, 544 (2003)
K. Symanzik, A modified model of Euclidean quantum field theory. Courant Institute of Mathematical Sciences, IMM-NYU 327 (1964)
K. Symanzik, Euclidean quantum field theory, in Local Quantum Field Theory, ed. by R. Jost (Academic Press, New York, 1968)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wipf, A. (2013). Scalar Fields at Zero and Finite Temperature. In: Statistical Approach to Quantum Field Theory. Lecture Notes in Physics, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33105-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-33105-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33104-6
Online ISBN: 978-3-642-33105-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)