C-symmetry

Charge conjugation for Dirac fields

The laws of electromagnetism (both classical and quantum) are invariant under the exchange of electrical charges with their negatives. For the case of electrons and quarks, both of which are fundamental particle fermion fields, the single-particle field excitations are described by the Dirac equation

${\displaystyle (i\mathrm{\partial }/-qA/-m)\psi =0}$

One wishes to find a charge-conjugate solution

${\displaystyle (i\mathrm{\partial }/+qA/-m){\psi }^c=0}$

A handful of algebraic manipulations are sufficient to obtain the second from the first. Standard expositions of the Dirac equation demonstrate a conjugate field ${\displaystyle \overline{\psi }={\psi }^{†}{\gamma }^0,}$ interpreted as an anti-particle field, satisfying the complex-transposed Dirac equation

${\displaystyle \overline{\psi }(-i\mathrm{\partial }/-qA/-m)=0}$

Note that some but not all of the signs have flipped. Transposing this back again gives almost the desired form, provided that one can find a 4×4 matrix ${\displaystyle C}$ that transposes the gamma matrices to insert the required sign-change:

${\displaystyle C^{-1}{\gamma }_{\mu }C=-{\gamma }_{\mu }^{\mathsf{\text{T}}}}$

The charge conjugate solution is then given by the involution

${\displaystyle \psi \mapsto {\psi }^c={\eta }_{c}C{\overline{\psi }}^{\mathsf{\text{T}}}}$

The 4×4 matrix ${\displaystyle C,}$ called the charge conjugation matrix, has an explicit form given in the article on gamma matrices. Curiously, this form is not representation-independent, but depends on the specific matrix representation chosen for the gamma group (the subgroup of the Clifford algebra capturing the algebraic properties of the gamma matrices). This matrix is representation dependent due to a subtle interplay involving the complexification of the spin group describing the Lorentz covariance of charged particles. The complex number ${\displaystyle {\eta }_c}$ is an arbitrary phase factor ${\displaystyle |{\eta }_c|=1,}$ generally taken to be ${\displaystyle {\eta }_c=1.}$

Charge conjugation, chirality, helicity

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

${\displaystyle C{\gamma }_5{C}^{-1}={\gamma }_5^{\mathsf{\text{T}}}}$

and whether or not ${\displaystyle {\gamma }_5^{\mathsf{\text{T}}}}$ equals ${\displaystyle {\gamma }_5}$ depends on the chosen representation for the gamma matrices. In the Dirac and chiral basis, one does have that ${\displaystyle {\gamma }_5^{\mathsf{\text{T}}}={\gamma }_5}$, while ${\displaystyle {\gamma }_5^{\mathsf{\text{T}}}=-{\gamma }_5}$ is obtained in the Majorana basis. A worked example follows.

Weyl spinors

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). One obtains this by writing the massless Dirac equation as

${\displaystyle i\mathrm{\partial }/\psi =0}$

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

${\displaystyle {{ϵ}_{ij}}^m{\sigma }^{ij}{\mathrm{\partial }}_m\psi ={\gamma }_5{\mathrm{\partial }}_t\psi }$

where ${\displaystyle {\sigma }^{\mu \nu }=i\left[{\gamma }^{\mu },{\gamma }^{\nu }\right]/2}$ is the angular momentum operator and ${\displaystyle {ϵ}_{ijk}}$ is the totally antisymmetric tensor. This can be brought to a slightly more recognizable form by defining the 3D spin operator ${\displaystyle {\mathrm{\Sigma }}^m\equiv {{ϵ}_{ij}}^m{\sigma }^{ij},}$ 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

${\displaystyle \left(\mathrm{\Sigma }\cdot \hat{k}\right)\psi ={\gamma }_5\psi .}$

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:

${\displaystyle \left(-p_0+\sigma \cdot \overrightarrow{p}\right){\psi }_{\text{R}}=0}$

and

${\displaystyle \left(p_0+\sigma \cdot \overrightarrow{p}\right){\psi }_{\text{L}}=0}$

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 ${\displaystyle p_{\mu }=i{\mathrm{\partial }}_{\mu }}$ is the momentum operator.

Charge conjugation in the chiral basis

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

${\displaystyle \psi =\left(\begin{array}{c}{\psi }_{\text{L}}\\ {\psi }_{\text{R}}\end{array}\right)}$

The corresponding dual (anti-particle) field is

${\displaystyle {\overline{\psi }}^{\mathsf{\text{T}}}={\left({\psi }^{†}{\gamma }^0\right)}^{\mathsf{\text{T}}}=\left(\begin{array}{cc}0& I\\ I& 0\end{array}\right)\left(\begin{array}{c}{\psi }_{\text{L}}^{\ast }\\ {\psi }_{\text{R}}^{\ast }\end{array}\right)=\left(\begin{array}{c}{\psi }_{\text{R}}^{\ast }\\ {\psi }_{\text{L}}^{\ast }\end{array}\right)}$

The charge-conjugate spinors are

${\displaystyle {\psi }^c=\left(\begin{array}{c}{\psi }_{\text{L}}^c\\ {\psi }_{\text{R}}^c\end{array}\right)={\eta }_{c}C{\overline{\psi }}^{\mathsf{\text{T}}}={\eta }_c\left(\begin{array}{cc}-i{\sigma }^2& 0\\ 0& i{\sigma }^2\end{array}\right)\left(\begin{array}{c}{\psi }_{\text{R}}^{\ast }\\ {\psi }_{\text{L}}^{\ast }\end{array}\right)={\eta }_c\left(\begin{array}{c}-i{\sigma }^2{\psi }_{\text{R}}^{\ast }\\ i{\sigma }^2{\psi }_{\text{L}}^{\ast }\end{array}\right)}$

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

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

Under combined charge and parity, one then has

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

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

Majorana condition

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

${\displaystyle {\psi }^{(+)}=\left(\begin{array}{c}{\psi }_{\text{L}}\\ i{\sigma }^2{\psi }_{\text{L}}^{\ast }\end{array}\right)}$

and

${\displaystyle {\psi }^{(-)}=\left(\begin{array}{c}i{\sigma }^2{\psi }_{\text{R}}^{\ast }\\ {\psi }_{\text{R}}\end{array}\right)}$

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 ${\displaystyle P^{(-)}=(1-\mathsf{C})P_{\text{R}}.}$

Geometric interpretation

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). 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.

Charge conjugation for quantized fields

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

1. ${\displaystyle \psi \mapsto {\psi }^c=\mathcal{C}\psi {\mathcal{C}}^{†}={\eta }_{c}C{\overline{\psi }}^{\mathsf{\text{T}}}}$
2. ${\displaystyle \overline{\psi }\mapsto {\overline{\psi }}^c=\mathcal{C}\overline{\psi }{\mathcal{C}}^{†}={\eta }_c^{\ast }{\psi }^{\mathsf{\text{T}}}{C}^{-1}}$
3. ${\displaystyle A_{\mu }\mapsto A_{\mu }^c=\mathcal{C}{A}_{\mu }{\mathcal{C}}^{†}=-A_{\mu }}$

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

Charge reversal in electroweak theory

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.

Scalar fields

The Dirac field has a "hidden" ${\displaystyle U(1)}$ gauge freedom, allowing it to couple directly to the electromagnetic field without any further modifications to the Dirac equation or the field itself. This is not the case for scalar fields, which must be explicitly "complexified" to couple to electromagnetism. This is done by "tensoring in" an additional factor of the complex plane ${\displaystyle \mathbb{C}}$ into the field, or constructing a Cartesian product with ${\displaystyle U(1)}$.

One very conventional technique is simply to start with two real scalar fields, ${\displaystyle \varphi }$ and ${\displaystyle \chi }$ and create a linear combination

${\displaystyle \psi \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{\varphi +i\chi }{\sqrt{2}}}$

The charge conjugation involution is then the mapping ${\displaystyle \mathsf{C}:i\mapsto -i}$ since this is sufficient to reverse the sign on the electromagnetic potential (since this complex number is being used to couple to it). For real scalar fields, charge conjugation is just the identity map: ${\displaystyle \mathsf{C}:\varphi \mapsto \varphi }$ and ${\displaystyle \mathsf{C}:\chi \mapsto \chi }$ and so, for the complexified field, charge conjugation is just ${\displaystyle \mathsf{C}:\psi \mapsto {\psi }^{\ast }.}$ The "mapsto" arrow ${\displaystyle \mapsto }$ is convenient for tracking "what goes where"; the equivalent older notation is simply to write ${\displaystyle \mathsf{C}\varphi =\varphi }$ and ${\displaystyle \mathsf{C}\chi =\chi }$ and ${\displaystyle \mathsf{C}\psi ={\psi }^{\ast }.}$

The above describes the conventional construction of a charged scalar field. It is also possible to introduce additional algebraic structure into the fields in other ways. In particular, one may define a "real" field behaving as ${\displaystyle \mathsf{C}:\varphi \mapsto -\varphi }$. As it is real, it cannot couple to electromagnetism by itself, but, when complexified, would result in a charged field that transforms as ${\displaystyle \mathsf{C}:\psi \mapsto -{\psi }^{\ast }.}$ Because C-symmetry is a discrete symmetry, one has some freedom to play these kinds of algebraic games in the search for a theory that correctly models some given physical reality.

In physics literature, a transformation such as ${\displaystyle \mathsf{C}:\varphi \mapsto {\varphi }^c=-\varphi }$ might be written without any further explanation. The formal mathematical interpretation of this is that the field ${\displaystyle \varphi }$ is an element of ${\displaystyle \mathbb{R}\times {\mathbb{Z}}_2}$ where ${\displaystyle {\mathbb{Z}}_2=\{+1,-1\}.}$ Thus, properly speaking, the field should be written as ${\displaystyle \varphi =(r,c)}$ which behaves under charge conjugation as ${\displaystyle \mathsf{C}:(r,c)\mapsto (r,-c).}$ It is very tempting, but not quite formally correct to just multiply these out, to move around the location of this minus sign; this mostly "just works", but a failure to track it properly will lead to confusion.

Combination of charge and parity reversal

It was believed for some time that C-symmetry could be combined with the parity-inversion transformation (see P-symmetry) to preserve a combined CP-symmetry. However, violations of this symmetry have been identified in the weak interactions (particularly in the kaons and B mesons). In the Standard Model, this CP violation is due to a single phase in the CKM matrix. If CP is combined with time reversal (T-symmetry), the resulting CPT-symmetry can be shown using only the Wightman axioms to be universally obeyed.

In general settings

The analog of charge conjugation can be defined for higher-dimensional gamma matrices, with an explicit construction for Weyl spinors given in the article on Weyl–Brauer matrices. Note, however, spinors as defined abstractly in the representation theory of Clifford algebras are not fields; rather, they should be thought of as existing on a zero-dimensional spacetime.

The analog of T-symmetry follows from ${\displaystyle {\gamma }^1{\gamma }^3}$ as the T-conjugation operator for Dirac spinors. Spinors also have an inherent P-symmetry, obtained by reversing the direction of all of the basis vectors of the Clifford algebra from which the spinors are constructed. The relationship to the P and T symmetries for a fermion field on a spacetime manifold are a bit subtle, but can be roughly characterized as follows. When a spinor is constructed via a Clifford algebra, the construction requires a vector space on which to build. By convention, this vector space is the tangent space of the spacetime manifold at a given, fixed spacetime point (a single fiber in the tangent manifold). P and T operations applied to the spacetime manifold can then be understood as also flipping the coordinates of the tangent space as well; thus, the two are glued together. Flipping the parity or the direction of time in one also flips it in the other. This is a convention. One can become unglued by failing to propagate this connection.

This is done by taking the tangent space as a vector space, extending it to a tensor algebra, and then using an inner product on the vector space to define a Clifford algebra. Treating each such algebra as a fiber, one obtains a fiber bundle called the Clifford bundle. Under a change of basis of the tangent space, elements of the Clifford algebra transform according to the spin group. Building a principle fiber bundle with the spin group as the fiber results in a spin structure.

All that is missing in the above paragraphs are the spinors themselves. These require the "complexification" of the tangent manifold: tensoring it with the complex plane. Once this is done, the Weyl spinors can be constructed. These have the form

${\displaystyle w_j=\frac{1}{\sqrt{2}}\left(e_{2j}-i{e}_{2j+1}\right)}$

where the ${\displaystyle e_j}$ are the basis vectors for the vector space ${\displaystyle V=T_{p}M}$, the tangent space at point ${\displaystyle p\in M}$ in the spacetime manifold ${\displaystyle M.}$ The Weyl spinors, together with their complex conjugates span the tangent space, in the sense that

${\displaystyle V\otimes \mathbb{C}=W\oplus \overline{W}}$

The alternating algebra ${\displaystyle \wedge W}$ is called the spinor space, it is where the spinors live, as well as products of spinors (thus, objects with higher spin values, including vectors and tensors).