C-symmetry

The charge conjugation symmetry is interpreted as that of electrical charge, because in all three cases (classical, quantum and geometry), one can construct Noether currents that resemble those of classical electrodynamics. This arises because electrodynamics itself, via Maxwell's equations, can be interpreted as a structure on a U(1) fiber bundle, the so-called circle bundle. This provides a geometric interpretation of electromagnetism: the electromagnetic potential ${\displaystyle A_{\mu }}$ is interpreted as the gauge connection (the Ehresmann connection) on the circle bundle. This geometric interpretation then allows (literally almost) anything possessing a complex-number-valued structure to be coupled to the electromagnetic field, provided that this coupling is done in a gauge-invariant way. Gauge symmetry, in this geometric setting, is a statement that, as one moves around on the circle, the coupled object must also transform in a "circular way", tracking in a corresponding fashion. More formally, one says that the equations must be gauge invariant under a change of local coordinate frames on the circle. For U(1), this is just the statement that the system is invariant under multiplication by a phase factor ${\displaystyle e^{i\phi (x)}}$ that depends on the (space-time) coordinate ${\displaystyle x.}$ In this geometric setting, charge conjugation can be understood as the discrete symmetry ${\displaystyle z=(x+iy)\mapsto {\overline {z}}=(x-iy)}$ that performs complex conjugation, that reverses the sense of direction around the circle.

In quantum field theory, charge conjugation can be understood as the exchange of particles with anti-particles. To understand this statement, one must have a minimal understanding of what quantum field theory is. In (vastly) simplified terms, it is a technique for performing calculations to obtain solutions for a system of coupled differential equations via perturbation theory. A key ingredient to this process is the quantum field, one for each of the (free, uncoupled) differential equations in the system. A quantum field is conventionally written as

${\displaystyle \psi (x)=\int d^{3}p\sum _{\sigma ,n}e^{-ip\cdot x}a\left({\vec {p}},\sigma ,n\right)u\left({\vec {p}},\sigma ,n\right)+e^{ip\cdot x}a^{\dagger }\left({\vec {p}},\sigma ,n\right)v\left({\vec {p}},\sigma ,n\right)}$

where ${\displaystyle {\vec {p}}}$ is the momentum, ${\displaystyle \sigma }$ is a spin label, ${\displaystyle n}$ is an auxiliary label for other states in the system. The ${\displaystyle a}$ and ${\displaystyle a^{\dagger }}$ are creation and annihilation operators (ladder operators) and ${\displaystyle u,v}$ are solutions to the (free, non-interacting, uncoupled) differential equation in question. The quantum field plays a central role because, in general, it is not known how to obtain exact solutions to the system of coupled differential questions. However, via perturbation theory, approximate solutions can be constructed as combinations of the free-field solutions. To perform this construction, one has to be able to extract and work with any one given free-field solution, on-demand, when required. The quantum field provides exactly this: it enumerates all possible free-field solutions in a vector space such that any one of them can be singled out at any given time, via the creation and annihilation operators.

The creation and annihilation operators obey the canonical commutation relations, in that the one operator "undoes" what the other "creates". This implies that any given solution ${\displaystyle u\left({\vec {p}},\sigma ,n\right)}$ must be paired with its "anti-solution" ${\displaystyle v\left({\vec {p}},\sigma ,n\right)}$ so that one undoes or cancels out the other. The pairing is to be performed so that all symmetries are preserved. As one is generally interested in Lorentz invariance, the quantum field contains an integral over all possible Lorentz coordinate frames, written above as an integral over all possible momenta (it is an integral over the fiber of the frame bundle). The pairing requires that a given ${\displaystyle u\left({\vec {p}}\right)}$ is associated with a ${\displaystyle v\left({\vec {p}}\right)}$ of the opposite momentum and energy. The quantum field is also a sum over all possible spin states; the dual pairing again matching opposite spins. Likewise for any other quantum numbers, these are also paired as opposites. There is a technical difficulty in carrying out this dual pairing: one must describe what it means for some given solution ${\displaystyle u}$ to be "dual to" some other solution ${\displaystyle v,}$ and to describe it in such a way that it remains consistently dual when integrating over the fiber of the frame bundle, when integrating (summing) over the fiber that describes the spin, and when integrating (summing) over any other fibers that occur in the theory.

When the fiber to be integrated over is the U(1) fiber of electromagnetism, the dual pairing is such that the direction (orientation) on the fiber is reversed. When the fiber to be integrated over is the SU(3) fiber of the color charge, the dual pairing again reverses orientation. This "just works" for SU(3) because it has two dual fundamental representations ${\displaystyle \mathbf {3} }$ and ${\displaystyle {\overline {\mathbf {3} }}}$ which can be naturally paired. This prescription for a quantum field naturally generalizes to any situation where one can enumerate the continuous symmetries of the system, and define duals in a coherent, consistent fashion. The pairing ties together opposite charges in the fully abstract sense. In physics, a charge is associated with a generator of a continuous symmetry. Different charges are associated with different eigenspaces of the Casimir invariants of the universal enveloping algebra for those symmetries. This is the case for both the Lorentz symmetry of the underlying spacetime manifold, as well as the symmetries of any fibers in the fiber bundle posed above the spacetime manifold. Duality replaces the generator of the symmetry with minus the generator. Charge conjugation is thus associated with reflection along the line bundle or determinant bundle of the space of symmetries.

The above then is a sketch of the general idea of a quantum field in quantum field theory. The physical interpretation is that solutions ${\displaystyle u\left({\vec {p}},\sigma ,n\right)}$ correspond to particles, and solutions ${\displaystyle v\left({\vec {p}},\sigma ,n\right)}$ correspond to antiparticles, and so charge conjugation is a pairing of the two. This sketch also provides enough hints to indicate what charge conjugation might look like in a general geometric setting. There is no particular forced requirement to use perturbation theory, to construct quantum fields that will act as middle-men in a perturbative expansion. Charge conjugation can be given a general setting.

For general Riemannian and pseudo-Riemannian manifolds, one has a tangent bundle, a cotangent bundle and a metric that ties the two together. There are several interesting things one can do, when presented with this situation. One is that the smooth structure allows differential equations to be posed on the manifold; the tangent and cotangent spaces provide enough structure to perform calculus on manifolds. Of key interest is the Laplacian, and, with a constant term, what amounts to the Klein–Gordon operator. Cotangent bundles, by their basic construction, are always symplectic manifolds. Symplectic manifolds have canonical coordinates ${\displaystyle x,p}$ interpreted as position and momentum, obeying canonical commutation relations. This provides the core infrastructure to extend duality, and thus charge conjugation, to this general setting.

A second interesting thing one can do is to construct a spin structure. Perhaps the most remarkable thing about this is that it is a very recognizable generalization to a ${\displaystyle (p,q)}$-dimensional pseudo-Riemannian manifold of the conventional physics concept of spinors living on a (1,3)-dimensional Minkowski spacetime. The construction passes through a complexified Clifford algebra to build a Clifford bundle and a spin manifold. At the end of this construction, one obtains a system that is remarkably familiar, if one is already acquainted with Dirac spinors and the Dirac equation. Several analogies pass through to this general case. First, the spinors are the Weyl spinors, and they come in complex-conjugate pairs. They are naturally anti-commuting (this follows from the Clifford algebra), which is exactly what one wants to make contact with the Pauli exclusion principle. Another is the existence of a chiral element, analogous to the gamma matrix ${\displaystyle \gamma _{5}}$ which sorts these spinors into left and right-handed subspaces. The complexification is a key ingredient, and it provides "electromagnetism" in this generalized setting. The spinor bundle doesn't "just" transform under the pseudo-orthogonal group ${\displaystyle SO(p,q)}$, the generalization of the Lorentz group ${\displaystyle SO(1,3)}$, but under a bigger group, the complexified spin group ${\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).}$ It is bigger in that it has a double covering by ${\displaystyle SO(p,q)\times U(1).}$

The ${\displaystyle U(1)}$ piece can be identified with electromagnetism in several different ways. One way is that the Dirac operators on the spin manifold, when squared, contain a piece ${\displaystyle F=dA}$ with ${\displaystyle A}$ arising from that part of the connection associated with the ${\displaystyle U(1)}$ piece. This is entirely analogous to what happens when one squares the ordinary Dirac equation in ordinary Minkowski spacetime. A second hint is that this ${\displaystyle U(1)}$ piece is associated with the determinant bundle of the spin structure, effectively tying together the left and right-handed spinors through complex conjugation.

What remains is to work through the discrete symmetries of the above construction. There are several that appear to generalize P-symmetry and T-symmetry. Identifying the ${\displaystyle p}$ dimensions with time, and the ${\displaystyle q}$ dimensions with space, one can reverse the tangent vectors in the ${\displaystyle p}$ dimensional subspace to get time reversal, and flipping the direction of the ${\displaystyle q}$ dimensions corresponds to parity. The C-symmetry can be identified with the reflection on the line bundle. To tie all of these together into a knot, one finally has the concept of transposition, in that elements of the Clifford algebra can be written in reversed (transposed) order. The net result is that not only do the conventional physics ideas of fields pass over to the general Riemannian setting, but also the ideas of the discrete symmetries.

There are two ways to react to this. One is to treat it as an interesting curiosity. The other is to realize that, in low dimensions (in low-dimensional spacetime) there are many "accidental" isomorphisms between various Lie groups and other assorted structures. Being able to examine them in a general setting disentangles these relationships, exposing more clearly "where things come from".

The interplay between chirality and charge conjugation is a bit subtle, and requires articulation. It is often said that charge conjugation does not alter the chirality of particles. This is not the case for fields, the difference arising in the "hole theory" interpretation of particles, where an anti-particle is interpreted as the absence of a particle. This is articulated below.

Conventionally, ${\displaystyle \gamma _{5}}$ is used as the chirality operator. Under charge conjugation, it transforms as

${\displaystyle C\gamma _{5}C^{-1}=\gamma _{5}^{\textsf {T}}}$

and whether or not ${\displaystyle \gamma _{5}^{\textsf {T}}}$ equals ${\displaystyle \gamma _{5}}$ depends on the chosen representation for the gamma matrices. In the Dirac and chiral basis, one does have that ${\displaystyle \gamma _{5}^{\textsf {T}}=\gamma _{5}}$, while ${\displaystyle \gamma _{5}^{\textsf {T}}=-\gamma _{5}}$ is obtained in the Majorana basis. A worked example follows.

For the case of massless Dirac spinor fields, chirality is equal to helicity for the positive energy solutions (and minus the helicity for negative energy solutions).[2]^{: § 2-4-3, page 87 ff} One obtains this by writing the massless Dirac equation as

${\displaystyle i\partial \!\!\!{\big /}\psi =0}$

Multiplying by ${\displaystyle \gamma ^{5}\gamma ^{0}=-i\gamma ^{1}\gamma ^{2}\gamma ^{3}}$ one obtains

${\displaystyle {\epsilon _{ij}}^{m}\sigma ^{ij}\partial _{m}\psi =\gamma _{5}\partial _{t}\psi }$

where ${\displaystyle \sigma ^{\mu \nu }=i\left[\gamma ^{\mu },\gamma ^{\nu }\right]/2}$ is the angular momentum operator and ${\displaystyle \epsilon _{ijk}}$ is the totally antisymmetric tensor. This can be brought to a slightly more recognizable form by defining the 3D spin operator ${\displaystyle \Sigma ^{m}\equiv {\epsilon _{ij}}^{m}\sigma ^{ij},}$ taking a plane-wave state ${\displaystyle \psi (x)=e^{-ik\cdot x}\psi (k)}$, applying the on-shell constraint that ${\displaystyle k\cdot k=0}$ and normalizing the momentum to be a 3D unit vector: ${\displaystyle {\hat {k}}_{i}=k_{i}/k_{0}}$ to write

${\displaystyle \left(\Sigma \cdot {\hat {k}}\right)\psi =\gamma _{5}\psi ~.}$

Examining the above, one concludes that angular momentum eigenstates (helicity eigenstates) correspond to eigenstates of the chiral operator. This allows the massless Dirac field to be cleanly split into a pair of Weyl spinors ${\displaystyle \psi _{\text{L}}}$ and ${\displaystyle \psi _{\text{R}},}$ each individually satisfying the Weyl equation, but with opposite energy:

${\displaystyle \left(-p_{0}+\sigma \cdot {\vec {p}}\right)\psi _{\text{R}}=0}$

and

${\displaystyle \left(p_{0}+\sigma \cdot {\vec {p}}\right)\psi _{\text{L}}=0}$

Note the freedom one has to equate negative helicity with negative energy, and thus the anti-particle with the particle of opposite helicity. To be clear, the ${\displaystyle \sigma }$ here are the Pauli matrices, and ${\displaystyle p_{\mu }=i\partial _{\mu }}$ is the momentum operator.

Taking the Weyl representation of the gamma matrices, one may write a (now taken to be massive) Dirac spinor as

${\displaystyle \psi ={\begin{pmatrix}\psi _{\text{L}}\\\psi _{\text{R}}\end{pmatrix}}}$

The corresponding dual (anti-particle) field is

${\displaystyle {\overline {\psi }}^{\textsf {T}}=\left(\psi ^{\dagger }\gamma ^{0}\right)^{\textsf {T}}={\begin{pmatrix}0&I\\I&0\end{pmatrix}}{\begin{pmatrix}\psi _{\text{L}}^{*}\\\psi _{\text{R}}^{*}\end{pmatrix}}={\begin{pmatrix}\psi _{\text{R}}^{*}\\\psi _{\text{L}}^{*}\end{pmatrix}}}$

The charge-conjugate spinors are

${\displaystyle \psi ^{c}={\begin{pmatrix}\psi _{\text{L}}^{c}\\\psi _{\text{R}}^{c}\end{pmatrix}}=\eta _{c}C{\overline {\psi }}^{\textsf {T}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}&0\\0&i\sigma ^{2}\end{pmatrix}}{\begin{pmatrix}\psi _{\text{R}}^{*}\\\psi _{\text{L}}^{*}\end{pmatrix}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}\psi _{\text{R}}^{*}\\i\sigma ^{2}\psi _{\text{L}}^{*}\end{pmatrix}}}$

where, as before, ${\displaystyle \eta _{c}}$ is a phase factor that can be taken to be ${\displaystyle \eta _{c}=1.}$ Note that the left and right states are inter-changed. This can be restored with a parity transformation. Under parity, the Dirac spinor transforms as

${\displaystyle \psi \left(t,{\vec {x}}\right)\mapsto \psi ^{p}\left(t,{\vec {x}}\right)=\gamma ^{0}\psi \left(t,-{\vec {x}}\right)}$

Under combined charge and parity, one then has

${\displaystyle \psi \left(t,{\vec {x}}\right)\mapsto \psi ^{cp}\left(t,{\vec {x}}\right)={\begin{pmatrix}\psi _{\text{L}}^{cp}\left(t,{\vec {x}}\right)\\\psi _{\text{R}}^{cp}\left(t,{\vec {x}}\right)\end{pmatrix}}=\eta _{c}{\begin{pmatrix}-i\sigma ^{2}\psi _{\text{L}}^{*}\left(t,-{\vec {x}}\right)\\i\sigma ^{2}\psi _{\text{R}}^{*}\left(t,-{\vec {x}}\right)\end{pmatrix}}}$

Conventionally, one takes ${\displaystyle \eta _{c}=1}$ globally. See however, the note below.

The Majorana condition imposes a constraint between the field and its charge conjugate, namely that they must be equal: ${\displaystyle \psi =\psi ^{c}.}$ This is perhaps best stated as the requirement that the Majorana spinor must be an eigenstate of the charge conjugation involution.

Doing so requires some notational care. In many texts discussing charge conjugation, the involution ${\displaystyle \psi \mapsto \psi ^{c}}$ is not given an explicit symbolic name, when applied to single-particle solutions of the Dirac equation. This is in contrast to the case when the quantized field is discussed, where a unitary operator ${\displaystyle {\mathcal {C}}}$ is defined (as done in a later section, below). For the present section, let the involution be named as ${\displaystyle {\mathsf {C}}:\psi \mapsto \psi ^{c}}$ so that ${\displaystyle {\mathsf {C}}\psi =\psi ^{c}.}$ Taking this to be a linear operator, one may consider its eigenstates. The Majorana condition singles out one such: ${\displaystyle {\mathsf {C}}\psi =\psi .}$ There are, however, two such eigenstates: ${\displaystyle {\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}.}$ Continuing in the Weyl basis, as above, these eigenstates are

${\displaystyle \psi ^{(+)}={\begin{pmatrix}\psi _{\text{L}}\\i\sigma ^{2}\psi _{\text{L}}^{*}\end{pmatrix}}}$

and

${\displaystyle \psi ^{(-)}={\begin{pmatrix}i\sigma ^{2}\psi _{\text{R}}^{*}\\\psi _{\text{R}}\end{pmatrix}}}$

The Majorana spinor is conventionally taken as just the positive eigenstate, namely ${\displaystyle \psi ^{(+)}.}$ The chiral operator ${\displaystyle \gamma _{5}}$ exchanges these two, in that

${\displaystyle \gamma _{5}{\mathsf {C}}=-{\mathsf {C}}\gamma _{5}}$

This is readily verified by direct substitution. Bear in mind that ${\displaystyle {\mathsf {C}}}$ does not have a 4×4 matrix representation! More precisely, there is no complex 4×4 matrix that can take a complex number to its complex conjugate; this inversion would require an 8×8 real matrix. The physical interpretation of complex conjugation as charge conjugation becomes clear when considering the complex conjugation of scalar fields, described in a subsequent section below.

The projectors onto the chiral eigenstates can be written as ${\displaystyle P_{\text{L}}=\left(1-\gamma _{5}\right)/2}$ and ${\displaystyle P_{\text{R}}=\left(1+\gamma _{5}\right)/2,}$ and so the above translates to

${\displaystyle P_{\text{L}}{\mathsf {C}}={\mathsf {C}}P_{\text{R}}~.}$

This directly demonstrates that charge conjugation, applied to single-particle complex-number-valued solutions of the Dirac equation flips the chirality of the solution. The projectors onto the charge conjugation eigenspaces are ${\displaystyle P^{(+)}=(1+{\mathsf {C}})P_{\text{L}}}$ and ${\displaystyle P^{(-)}=(1-{\mathsf {C}})P_{\text{R}}.}$

The phase factor ${\displaystyle \ \eta _{c}\ }$ can be given a geometric interpretation. It has been noted that, for massive Dirac spinors, the "arbitrary" phase factor ${\displaystyle \ \eta _{c}\ }$ may depend on both the momentum, and the helicity (but not the chirality).[4] This can be interpreted as saying that this phase may vary along the fiber of the spinor bundle, depending on the local choice of a coordinate frame. Put another way, a spinor field is a local section of the spinor bundle, and Lorentz boosts and rotations correspond to movements along the fibers of the corresponding frame bundle (again, just a choice of local coordinate frame). Examined in this way, this extra phase freedom can be interpreted as the phase arising from the electromagnetic field. For the Majorana spinors, the phase would be constrained to not vary under boosts and rotations.

The above describes charge conjugation for the single-particle solutions only. When the Dirac field is second-quantized, as in quantum field theory, the spinor and electromagnetic fields are described by operators. The charge conjugation involution then manifests as a unitary operator ${\displaystyle {\mathcal {C}}}$ (in calligraphic font) acting on the particle fields, expressed as[5][6]

1. ${\displaystyle \psi \mapsto \psi ^{c}={\mathcal {C}}\ \psi \ {\mathcal {C}}^{\dagger }=\eta _{c}\ C\ {\overline {\psi }}^{\textsf {T}}}$
2. ${\displaystyle {\overline {\psi }}\mapsto {\overline {\psi }}^{c}={\mathcal {C}}\ {\overline {\psi }}\ {\mathcal {C}}^{\dagger }=\eta _{c}^{*}\ \psi ^{\textsf {T}}\ C^{-1}}$
3. ${\displaystyle A_{\mu }\mapsto A_{\mu }^{c}={\mathcal {C}}\ A_{\mu }\ {\mathcal {C}}^{\dagger }=-A_{\mu }\ }$

where the non-calligraphic ${\displaystyle \ C\ }$ is the same 4×4 matrix given before.

Charge conjugation does not alter the chirality of particles. A left-handed neutrino would be taken by charge conjugation into a left-handed antineutrino, which does not interact in the Standard Model. This property is what is meant by the "maximal violation" of C-symmetry in the weak interaction.

Some postulated extensions of the Standard Model, like left-right models, restore this C-symmetry.