44 Symmetries and Relativistic Scalar Quantum Fields

Just as for non-relativistic quantum fields, the theory of free relativistic scalar quantum fields starts by taking as phase space an infinite dimensional space of solutions of an equation of motion. Quantization of this phase space proceeds by constructing field operators which provide a representation of the corresponding Heisenberg Lie algebra, using an infinite dimensional version of the Bargmann-Fock construction. In both cases the equation of motion has a representationtheoretical significance: it is an eigenvalue equation for the Casimir operator of a group of space-time symmetries, picking out an irreducible representation of that group. In the non-relativistic case, the Laplacian $\Delta$ was the Casimir operator, the symmetry group was the Euclidean group $E(3)$ and one got an irreducible representation for fixed energy. In the relativistic case the Casimir operator is the Minkowski space version of the Laplacian

$$-\frac{{\partial }^2}{\partial t^2}+\Delta$$

the space-time symmetry group is the Poincar´e group, and the eigenvalue of the Casimir is $m^2$

The Poincar´e group acts on the phase space of solutions to the Klein-Gordon equation, preserving the Poisson bracket. The same general methods as in the finite dimensional and non-relativistic quantum field theory cases can be used to get a representation of the Poincar´e group by intertwining operators for the Heisenberg Lie algebra representation (the representation given by the field operators). These methods give a representation of the Lie algebra of the Poincar´e group in terms of quadratic combinations of the field operators.

We’ll begin though with the case of an even simpler group action on the phase space, that coming from an “internal symmetry” of multi-component scalar fields, with an orthogonal group or unitary group acting on the real or complex vector space in which the classical fields take their values. For the simplest case of fields taking values in ${\mathbf{R}}^2$ or $\mathbf{C},$ one gets a theory of charged relativistic particles, with antiparticles now distinguishable from particles (they have opposite charge).

中文目录

Chapter contents

• 44.1 Internal symmetries
  • 44.1.1 $SO(m)$ symmetry and real scalar fields
  • 44.1.2 U(1) symmetry and complex scalar fields
• 44.2 Poincar´e symmetry and scalar fields
  • 44.2.1 Translations
  • 44.2.2 Rotations
  • 44.2.3 Boosts
• 44.3 For further reading

Previous chapter: 43 The Klein-Gordon Equation and Scalar Quantum Fields

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来源版本：2025-10-20

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