Parity and Interactions: Misreading or Misunderstanding?

[KevinMcHugh]

TL;DR Summary

Is intrinsic parity meaningless for free particles

I read in Greiner Field Quantization that parity only has meaning in interactions. Did I misread or not understand something correctly?

 

[vanhees71]

If I remember right, that refers to the notion of "intrinsic parity" of scalar vs. pseudoscalar operators. For free fields it indeed doesn't matter much, but if it comes to interactions it can become important for model building, i.e., when trying to construct a Lagrangian describing particles and their interactions.

E.g., today we know that Dirac must have discovered his equation when thinking about electrons and electromagnetic interactions to build a relativistic theory for atomic physics. First of all one knew an electron is a massive particle with spin 1/2, and the electromagnetic interactions are invariant under parity and charge conjugation. The electromagnetic interaction itself is described as the electromagnetic field. What Dirac did not yet know in 1927 was the representation theory of the Poincare group.

Knowing this the arguments are pretty simple: One starts with the proper orthochronous Poincare group, which must be strictly valid as a symmetry group in order to have a relativistic theory, because the space-time model, Minkowski space, obeys this symmetry. An analysis of the unitary representations reveals that there are two big classes: massless and massive representations. Since electrons are massive representations, we first discuss them. We start with free, non-interacting particles. A complete one-particle basis is, according to the representation theory of the Poincare group, described by a value for the mass (the mass is one of the Casimir operators, classifying the irreducible unitary representations). Now one can always boost to a frame, where the particle is at rest, and the only remaining symmetry for our one-particle energy-momentum eigenstate is the rotation subgroup, keeping the momentum of the particle in its restframe 0. This "little group" for the massive case has well known representations, and this describes the spin, i.e., the intrinsic angular momentum of the particle. An electron has spin 1/2, and thus the electron at rest has two spin states with $\sigma_z \in \{1/2,-1/2 \}$ (using natural units with $\hbar=1$). As it turns out, when extending this SU(2) representation of the rotation group to the full proper orthochronous Poincare group, which is a unique procedure, it turns out there are two distinct representations, labelled (1/2,0) and (0,1/2).

Now to decide which representation to use for the electrons, parity comes into the game. It turns out that neither with the (1/2,0) nor the (0,1/2) representation can we describe parity, i.e., the behavior of the states under spatial reflections other than trivially. It turns out that we need a direct sum of both representations, and the two-component spinors are lumped together to a four-dimensional quantity called Dirac spinor, consisting of two usual two-dimensional spinors. Under space reflections these spinors within the Dirac spinor are simply exchanged. Now in some way we have defined a chirality, i.e., if you call one Weyl-spinor part of Dirac "left handed" (it doesn't matter which of the two you call "left handed") under space reflection it's exchanged with the other Weyl-spinor, which you thus call "right handed".

It now also turns out that these "doubling" of field-degrees of freedom comes in handy for constructing a local QFT with a stable ground state. As is well known, the equation of motion, which just says that $E^3=\vec{p}^2+m^2$ when applied to energy-momentum eigenstates, there's the trouble that the complete Fourier space includes positive- and negative-frequency modes, and that this can be easily remedied with the notion that energy should be bounded from below, when quantizing the corresponding fields: One simply writes an annihilation operator in front of the positive-frequency modes and a creation operator in front of the negative-frequency modes. Then all single-particle states have positive energy (which is true for all many-body states either since for non-interacting particles the total energy is simply the sum of the single-particle energies). In order to get in addition a local theory, i.e., where the field operators transform in the usual way as their classical analogues under Poincare transformations, one has to quantize these spin-1/2 particles as fermions, i.e., with anti-commutation relations for the creation and annihilation operators.

The only question you have to decide now is whether the creation operators in front of the negative-frequency modes of the field operator is the adjoint of the annihilation operators in front of the positive-frequency modes or not. As it turns out in order to also have a conserved charge you need to make it a different creation operator. So the theory of free electrons now has forced us to introduce a second sort of particles with precisely the same mass and, as the Noether analysis of symmetry under global phaseredefinitions of the field operators shows, opposite charge quantum numbers. Thus despite an electron we have another particle with exactly the same mass but opposite charge in the game. That's called and antielectron or positron since we want to make this charge now the electric charge and for historical reasons the electrons carry a negative charge, so that the antielectrons must have a positive charge of the same magnitude.

Now one brings in the interactions to describe electromagnetism. As is well known since Maxwell's times, this is described by the electromagnetic field which in the analysis in terms of the Poincare group is a massless spin-1 field, and the corresponding analysis of the representation theory of the Poincare group tells us that this must be described as a gauge field, because otherwise one would have continuously many spin-like degrees of freedom coming with the "particles" described by such a massless spin-1 field, but this we cannot use, because there's no hint for such a thing in any observations.

This now makes life much easier, because since Yang and Mills we know, how to make a nice gauge theory out of the free-particle Dirac theory. Since we have only one electromagnetic field, it's just telling us to "gauge the symmetry" under multiplication of the Dirac spinor field with a phase factor, i.e., to make it valid as a local symmetry, which let's us naturally introduce the electromagnetic field as the gauge-boson of this local symmetry.

Finally we have to decide which interactions terms are allowed by all the symmetries, and there parity comes into the game again. We also like to have a renormalizable theory, though this is nowadays not such a hard demand anymore since we've learned that in some sense all QFTs are effective theories valid at a limited range of energies. Neverthess all that's left is the well-known QED Lagrangian with spin-1/2 matter fields, in this most simple form describing electrons, positrons, and photons.

All concerning parity within this most simple example for a practically important QFT, which is not just convention is the fact that particle and antiparticle states have necessarily opposite parity. E.g., positronium in its $s$ state (i.e., angular momentum 0, parapositronium) has negative parity.

 

[KevinMcHugh]

If I remember right, that refers to the notion of "intrinsic parity" of scalar vs. pseudoscalar operators. For free fields it indeed doesn't matter much, but if it comes to interactions it can become important for model building, i.e., when trying to construct a Lagrangian describing particles and their interactions.

E.g., today we know that Dirac must have discovered his equation when thinking about electrons and electromagnetic interactions to build a relativistic theory for atomic physics. First of all one knew an electron is a massive particle with spin 1/2, and the electromagnetic interactions are invariant under parity and charge conjugation. The electromagnetic interaction itself is described as the electromagnetic field. What Dirac did not yet know in 1927 was the representation theory of the Poincare group.

Knowing this the arguments are pretty simple: One starts with the proper orthochronous Poincare group, which must be strictly valid as a symmetry group in order to have a relativistic theory, because the space-time model, Minkowski space, obeys this symmetry. An analysis of the unitary representations reveals that there are two big classes: massless and massive representations. Since electrons are massive representations, we first discuss them. We start with free, non-interacting particles. A complete one-particle basis is, according to the representation theory of the Poincare group, described by a value for the mass (the mass is one of the Casimir operators, classifying the irreducible unitary representations). Now one can always boost to a frame, where the particle is at rest, and the only remaining symmetry for our one-particle energy-momentum eigenstate is the rotation subgroup, keeping the momentum of the particle in its restframe 0. This "little group" for the massive case has well known representations, and this describes the spin, i.e., the intrinsic angular momentum of the particle. An electron has spin 1/2, and thus the electron at rest has two spin states with $\sigma_z \in \{1/2,-1/2 \}$ (using natural units with $\hbar=1$). As it turns out, when extending this SU(2) representation of the rotation group to the full proper orthochronous Poincare group, which is a unique procedure, it turns out there are two distinct representations, labelled (1/2,0) and (0,1/2).

Now to decide which representation to use for the electrons, parity comes into the game. It turns out that neither with the (1/2,0) nor the (0,1/2) representation can we describe parity, i.e., the behavior of the states under spatial reflections other than trivially. It turns out that we need a direct sum of both representations, and the two-component spinors are lumped together to a four-dimensional quantity called Dirac spinor, consisting of two usual two-dimensional spinors. Under space reflections these spinors within the Dirac spinor are simply exchanged. Now in some way we have defined a chirality, i.e., if you call one Weyl-spinor part of Dirac "left handed" (it doesn't matter which of the two you call "left handed") under space reflection it's exchanged with the other Weyl-spinor, which you thus call "right handed".

It now also turns out that these "doubling" of field-degrees of freedom comes in handy for constructing a local QFT with a stable ground state. As is well known, the equation of motion, which just says that $E^3=\vec{p}^2+m^2$ when applied to energy-momentum eigenstates, there's the trouble that the complete Fourier space includes positive- and negative-frequency modes, and that this can be easily remedied with the notion that energy should be bounded from below, when quantizing the corresponding fields: One simply writes an annihilation operator in front of the positive-frequency modes and a creation operator in front of the negative-frequency modes. Then all single-particle states have positive energy (which is true for all many-body states either since for non-interacting particles the total energy is simply the sum of the single-particle energies). In order to get in addition a local theory, i.e., where the field operators transform in the usual way as their classical analogues under Poincare transformations, one has to quantize these spin-1/2 particles as fermions, i.e., with anti-commutation relations for the creation and annihilation operators.

The only question you have to decide now is whether the creation operators in front of the negative-frequency modes of the field operator is the adjoint of the annihilation operators in front of the positive-frequency modes or not. As it turns out in order to also have a conserved charge you need to make it a different creation operator. So the theory of free electrons now has forced us to introduce a second sort of particles with precisely the same mass and, as the Noether analysis of symmetry under global phaseredefinitions of the field operators shows, opposite charge quantum numbers. Thus despite an electron we have another particle with exactly the same mass but opposite charge in the game. That's called and antielectron or positron since we want to make this charge now the electric charge and for historical reasons the electrons carry a negative charge, so that the antielectrons must have a positive charge of the same magnitude.

Now one brings in the interactions to describe electromagnetism. As is well known since Maxwell's times, this is described by the electromagnetic field which in the analysis in terms of the Poincare group is a massless spin-1 field, and the corresponding analysis of the representation theory of the Poincare group tells us that this must be described as a gauge field, because otherwise one would have continuously many spin-like degrees of freedom coming with the "particles" described by such a massless spin-1 field, but this we cannot use, because there's no hint for such a thing in any observations.

This now makes life much easier, because since Yang and Mills we know, how to make a nice gauge theory out of the free-particle Dirac theory. Since we have only one electromagnetic field, it's just telling us to "gauge the symmetry" under multiplication of the Dirac spinor field with a phase factor, i.e., to make it valid as a local symmetry, which let's us naturally introduce the electromagnetic field as the gauge-boson of this local symmetry.

Finally we have to decide which interactions terms are allowed by all the symmetries, and there parity comes into the game again. We also like to have a renormalizable theory, though this is nowadays not such a hard demand anymore since we've learned that in some sense all QFTs are effective theories valid at a limited range of energies. Neverthess all that's left is the well-known QED Lagrangian with spin-1/2 matter fields, in this most simple form describing electrons, positrons, and photons.

All concerning parity within this most simple example for a practically important QFT, which is not just convention is the fact that particle and antiparticle states have necessarily opposite parity. E.g., positronium in its $s$ state (i.e., angular momentum 0, parapositronium) has negative parity.

Sir, thank you for that detailed answer. It clarified a couple of things for me. And your memory is correct, intrinsic parity for scalar operators.