134
20 Constructive Symmetry Breaking
where ϕ is a real scalar field transforming as an n-dimensional irreducible representation of the internal symmetry group O(n). The Goldstone strategy is based on the
following steps:
i) (semi-classical approximation) one considers the classical absolute minima ϕ min
of the (classical) potential U (which form an orbit under O(n))
ii) (perturbative expansion about the mean field semi-classical approximation) one
picks up one absolute minimum ϕ min and builds up a perturbative quantum
expansion around such a classical value of the field: ϕ = ϕ min + χ.
The expansion is conveniently organized as a quantum (or loop) expansion in .
111
It is an important result that such an expansion makes sense, namely that a renormalized perturbation expansion exists.
112 It then follows that in such a perturbative
expansion
< ϕ > 0 = ϕ min + small quantum corrections
and therefore, if ϕ min is not symmetric, so is < ϕ > 0 . In this way, one constructs a
(perturbative) theory with a symmetry breaking order parameter. By this logic, each
absolute minimum identifies a ground state and a non-symmetric theory.
B. Ruelle Non-perturbative Strategy
The Goldstone strategy has proved successful for the application to the many-body
theory (e.g. the Ginzburg–Landau model of superconductivity) and for elementary
particle theory (see the perturbative treatment of the standard model of electromagnetic and weak interactions), but it leaves some basic questions open. In fact, it is
known that mean field approximations are often not reliable and the results on the
triviality of the ϕ
4 theory in four space-time dimensions seem to indicate that the
perturbative expansion, which might be, at best, an asymptotic expansion, may have
little to do with the non-perturbative solution.
A strategy for a non-perturbative approach to symmetry breaking in quantum field
theory and in the many-body theory is provided by the imaginary time (or Euclidean)
formulation and the functional integral representation of the Euclidean correlation
functions.
113
111 S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973).
112 B.W. Lee, Nucl. Phys. B9, 649 (1969); K. Symanzik, Renormalization of Theories with Broken
Symmetry, in Cargèse Lectures in Physics 1970, D. Bessis ed., Gordon and Breach, New York
1972; C. Becchi, A. Rouet and R. Stora, Renormalizable Theories with Symmetry Breaking, in
Field Theory, Quantization and Statistical Physics, E. Tirapegui ed., D. Reidel 1981. For textbook
accounts, see, e.g. J. Collins, Renormalization, Cambridge Univ. Press 1984, Chap. 9; L.S. Brown,
Quantum Field Theory, Cambridge Univ. Press 1994.
113 See J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd ed.,
Springer 1987. For a handy account, see [SNS 96].
20 Constructive Symmetry Breaking
where ϕ is a real scalar field transforming as an n-dimensional irreducible representation of the internal symmetry group O(n). The Goldstone strategy is based on the
following steps:
i) (semi-classical approximation) one considers the classical absolute minima ϕ min
of the (classical) potential U (which form an orbit under O(n))
ii) (perturbative expansion about the mean field semi-classical approximation) one
picks up one absolute minimum ϕ min and builds up a perturbative quantum
expansion around such a classical value of the field: ϕ = ϕ min + χ.
The expansion is conveniently organized as a quantum (or loop) expansion in .
111
It is an important result that such an expansion makes sense, namely that a renormalized perturbation expansion exists.
112 It then follows that in such a perturbative
expansion
< ϕ > 0 = ϕ min + small quantum corrections
and therefore, if ϕ min is not symmetric, so is < ϕ > 0 . In this way, one constructs a
(perturbative) theory with a symmetry breaking order parameter. By this logic, each
absolute minimum identifies a ground state and a non-symmetric theory.
B. Ruelle Non-perturbative Strategy
The Goldstone strategy has proved successful for the application to the many-body
theory (e.g. the Ginzburg–Landau model of superconductivity) and for elementary
particle theory (see the perturbative treatment of the standard model of electromagnetic and weak interactions), but it leaves some basic questions open. In fact, it is
known that mean field approximations are often not reliable and the results on the
triviality of the ϕ
4 theory in four space-time dimensions seem to indicate that the
perturbative expansion, which might be, at best, an asymptotic expansion, may have
little to do with the non-perturbative solution.
A strategy for a non-perturbative approach to symmetry breaking in quantum field
theory and in the many-body theory is provided by the imaginary time (or Euclidean)
formulation and the functional integral representation of the Euclidean correlation
functions.
113
111 S. Coleman and E. Weinberg, Phys. Rev. D7, 1888 (1973).
112 B.W. Lee, Nucl. Phys. B9, 649 (1969); K. Symanzik, Renormalization of Theories with Broken
Symmetry, in Cargèse Lectures in Physics 1970, D. Bessis ed., Gordon and Breach, New York
1972; C. Becchi, A. Rouet and R. Stora, Renormalizable Theories with Symmetry Breaking, in
Field Theory, Quantization and Statistical Physics, E. Tirapegui ed., D. Reidel 1981. For textbook
accounts, see, e.g. J. Collins, Renormalization, Cambridge Univ. Press 1984, Chap. 9; L.S. Brown,
Quantum Field Theory, Cambridge Univ. Press 1994.
113 See J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd ed.,
Springer 1987. For a handy account, see [SNS 96].