Gamma matrices
In mathematical physics, the gamma matrices, {γ0,γ1,γ2,γ3}{displaystyle {gamma ^{0},gamma ^{1},gamma ^{2},gamma ^{3}}}, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3(R). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles.
In Dirac representation, the four contravariant gamma matrices are
- γ0=(1000010000−10000−1),γ1=(000100100−100−1000),γ2=(000−i00i00i00−i000),γ3=(0010000−1−10000100).{displaystyle {begin{aligned}gamma ^{0}&={begin{pmatrix}1&0&0&0\0&1&0&0\0&0&-1&0\0&0&0&-1end{pmatrix}},&gamma ^{1}&={begin{pmatrix}0&0&0&1\0&0&1&0\0&-1&0&0\-1&0&0&0end{pmatrix}},\gamma ^{2}&={begin{pmatrix}0&0&0&-i\0&0&i&0\0&i&0&0\-i&0&0&0end{pmatrix}},&gamma ^{3}&={begin{pmatrix}0&0&1&0\0&0&0&-1\-1&0&0&0\0&1&0&0end{pmatrix}}.end{aligned}}}
γ0{displaystyle gamma ^{0}} is the time-like, hermitian matrix. The other three are space-like, antihermitian matrices. More compactly, γ0=σ3⊗I{displaystyle gamma ^{0}=sigma ^{3}otimes I}, and γi=iσ2⊗σi{displaystyle gamma ^{i}=isigma ^{2}otimes sigma ^{i}}.
Analogous sets of gamma matrices can be defined in any dimension and for any signature of the metric. For example, the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3, 0). In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma-matrix to be presented below generate the Clifford algebra.
Contents
1 Mathematical structure
2 Physical structure
3 Expressing the Dirac equation
4 The fifth gamma matrix, γ5
5 Identities
5.1 Miscellaneous identities
5.2 Trace identities
5.3 Normalization
5.4 Feynman slash notation used in Quantum field theory
6 Other representations
6.1 Dirac basis
6.2 Weyl (chiral) basis
6.3 Majorana basis
6.4 Cℓ1,3(C) and Cℓ1,3(R)
7 Euclidean Dirac matrices
7.1 Chiral representation
7.2 Non-relativistic representation
8 See also
9 References
10 External links
Mathematical structure
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation
- {γμ,γν}=γμγν+γνγμ=2ημνI4,{displaystyle {gamma ^{mu },gamma ^{nu }}=gamma ^{mu }gamma ^{nu }+gamma ^{nu }gamma ^{mu }=2eta ^{mu nu }I_{4},}
where {,}{displaystyle {,}} is the anticommutator, ημν{displaystyle eta ^{mu nu }} is the Minkowski metric with signature (+ − − −), and I4{displaystyle I_{4}} is the 4 × 4 identity matrix.
This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices. Covariant gamma matrices are defined by
- γμ=ημνγν={γ0,−γ1,−γ2,−γ3},{displaystyle gamma _{mu }=eta _{mu nu }gamma ^{nu }=left{gamma ^{0},-gamma ^{1},-gamma ^{2},-gamma ^{3}right},}
and Einstein notation is assumed.
Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation:
- {γμ,γν}=−2ημνI4{displaystyle {gamma ^{mu },gamma ^{nu }}=-2eta ^{mu nu }I_{4}}
or a multiplication of all gamma matrices by i{displaystyle i}, which of course changes their hermiticity properties detailed below. Under the alternative sign convention for the metric the covariant gamma matrices are then defined by
- γμ=ημνγν={−γ0,+γ1,+γ2,+γ3}.{displaystyle gamma _{mu }=eta _{mu nu }gamma ^{nu }=left{-gamma ^{0},+gamma ^{1},+gamma ^{2},+gamma ^{3}right}.}
Physical structure
The Clifford algebra Cl1,3(R) over spacetime V can be regarded as the set of real linear operators from V to itself, End(V), or more generally, when complexified to Cl1,3(R)C, as the set of linear operators from any 4-dimensional complex vector space to itself. More simply, given a basis for V, Cl1,3(R)C is just the set of all 4 × 4 complex matrices, but endowed with a Clifford algebra structure. Spacetime is assumed to be endowed with the Minkowski metric ημν. A space of bispinors, Ux, is also assumed at every point in spacetime, endowed with the bispinor representation of the Lorentz group. The bispinor fields Ψ of the Dirac equations, evaluated at any point x in spacetime, are elements of Ux, see below. The Clifford algebra is assumed to act on Ux as well (by matrix multiplication with column vectors Ψ(x) in Ux for all x). This will be the primary view of elements of Cl1,3(R)C in this section.
For each linear transformation S of Ux, there is a transformation of End(Ux) given by SES−1 for E in Cl1,3(R)C ≈ End(Ux). If S belongs to a representation of the Lorentz group, then the induced action E ↦ SES−1 will also belong to a representation of the Lorentz group, see Representation theory of the Lorentz group.
If S(Λ) is the bispinor representation acting on Ux of an arbitrary Lorentz transformation Λ in the standard (4-vector) representation acting on V, then there is a corresponding operator on End(Ux) = Cl1,3(R)C given by
- γμ↦S(Λ)γμS(Λ)−1=(Λ−1)μνγν:=Λνμγν,{displaystyle gamma ^{mu }mapsto S(Lambda )gamma ^{mu }S(Lambda )^{-1}={(Lambda ^{-1})^{mu }}_{nu }gamma ^{nu }:={Lambda _{nu }}^{mu }gamma ^{nu },}
showing that the γμ can be viewed as a basis of a representation space of the 4-vector representation of the Lorentz group sitting inside the Clifford algebra. This means that quantities of the form
- a/:=aμγμ{displaystyle a!!!/:=a_{mu }gamma ^{mu }}
should be treated as 4-vectors in manipulations. It also means that indices can be raised and lowered on the γ using the metric ημν as with any 4-vector. The notation is called the Feynman slash notation. The slash operation maps the basis eμ of V, or any 4-dimensional vector space, to basis vectors γμ. The transformation rule for slashed quantities is simply
- a/μ↦Λμνa/ν.{displaystyle {a!!!/}^{mu }mapsto {Lambda ^{mu }}_{nu }{a!!!/}^{nu }.}
One should note that this is different from the transformation rule for the γμ, which are now treated as (fixed) basis vectors. The designation of the 4-tuple (γμ) = (γ0, γ1, γ2, γ3) as a 4-vector sometimes found in the literature is thus a slight misnomer. The latter transformation corresponds to an active transformation of the components of a slashed quantity in terms of the basis γμ, and the former to a passive transformation of the basis γμ itself.
The elements σμν = γμγν − γνγμ form a representation of the Lie algebra of the Lorentz group. This is a spin representation. When these matrices, and linear combinations of them, are exponentiated, they are bispinor representations of the Lorentz group, e.g., the S(Λ) of above are of this form. The 6-dimensional space the σμν span is the representation space of a tensor representation of the Lorentz group. For the higher order elements of the Clifford algebra in general and their transformation rules, see the article Dirac algebra. But it is noted here that the Clifford algebra has no subspace being the representation space of a spin representation of the Lorentz group in the context used here.
Expressing the Dirac equation
In natural units, the Dirac equation may be written as
- (iγμ∂μ−m)ψ=0{displaystyle (igamma ^{mu }partial _{mu }-m)psi =0}
where ψ{displaystyle psi } is a Dirac spinor.
Switching to Feynman notation, the Dirac equation is
- (i∂/−m)ψ=0.{displaystyle (i{partial !!!/}-m)psi =0.}
The fifth gamma matrix, γ5
It is useful to define the product of the four gamma matrices as follows:
γ5:=iγ0γ1γ2γ3=(0010000110000100){displaystyle gamma ^{5}:=igamma ^{0}gamma ^{1}gamma ^{2}gamma ^{3}={begin{pmatrix}0&0&1&0\0&0&0&1\1&0&0&0\0&1&0&0end{pmatrix}}} (in the Dirac basis).
Although γ5{displaystyle gamma ^{5}} uses the letter gamma, it is not one of the gamma matrices of Cℓ1,3(R). The number 5 is a relic of old notation in which γ0{displaystyle gamma ^{0}} was called "γ4{displaystyle gamma ^{4}}".
γ5{displaystyle gamma ^{5}} has also an alternative form:
- γ5=i4!εμναβγμγνγαγβ{displaystyle gamma ^{5}={frac {i}{4!}}varepsilon _{mu nu alpha beta }gamma ^{mu }gamma ^{nu }gamma ^{alpha }gamma ^{beta }}
This can be seen by exploiting the fact that all the four gamma matrices anticommute, so
γ0γ1γ2γ3=γ[0γ1γ2γ3]=14!δμνϱσ0123γμγνγϱγσ{displaystyle gamma ^{0}gamma ^{1}gamma ^{2}gamma ^{3}=gamma ^{[0}gamma ^{1}gamma ^{2}gamma ^{3]}={frac {1}{4!}}delta _{mu nu varrho sigma }^{0123}gamma ^{mu }gamma ^{nu }gamma ^{varrho }gamma ^{sigma }},
where δμνϱσαβγδ{displaystyle delta _{mu nu varrho sigma }^{alpha beta gamma delta }} is the type (4,4) generalized Kronecker delta in 4 dimensions, in full antisymmetrization. If εα…β{displaystyle varepsilon _{alpha dots beta }} denotes the Levi-Civita symbol in n dimensions, we can use the identity δμνϱσαβγδ=εαβγδεμνϱσ{displaystyle delta _{mu nu varrho sigma }^{alpha beta gamma delta }=varepsilon ^{alpha beta gamma delta }varepsilon _{mu nu varrho sigma }}.
Then we get
- γ5=iγ0γ1γ2γ3=i4!ε0123εμνϱσγμγνγϱγσ=i4!εμνϱσγμγνγϱγσ{displaystyle gamma ^{5}=igamma ^{0}gamma ^{1}gamma ^{2}gamma ^{3}={frac {i}{4!}}varepsilon ^{0123}varepsilon _{mu nu varrho sigma },gamma ^{mu }gamma ^{nu }gamma ^{varrho }gamma ^{sigma }={frac {i}{4!}}varepsilon _{mu nu varrho sigma },gamma ^{mu }gamma ^{nu }gamma ^{varrho }gamma ^{sigma }}
This matrix is useful in discussions of quantum mechanical chirality. For example, a Dirac field can be projected onto its left-handed and right-handed components by:
ψL=1−γ52ψ,ψR=1+γ52ψ{displaystyle psi _{L}={frac {1-gamma ^{5}}{2}}psi ,qquad psi _{R}={frac {1+gamma ^{5}}{2}}psi }.
Some properties are:
- It is hermitian:
- (γ5)†=γ5.{displaystyle (gamma ^{5})^{dagger }=gamma ^{5}.}
- Its eigenvalues are ±1, because:
- (γ5)2=I4.{displaystyle (gamma ^{5})^{2}=I_{4}.}
- It anticommutes with the four gamma matrices:
- {γ5,γμ}=γ5γμ+γμγ5=0.{displaystyle left{gamma ^{5},gamma ^{mu }right}=gamma ^{5}gamma ^{mu }+gamma ^{mu }gamma ^{5}=0.}
The set {γ0, γ1, γ2, γ3, iγ5} therefore, by the last two properties (keeping in mind that i2 = −1) and those of the old gammas, forms the basis of the Clifford algebra in 5 spacetime dimensions for the metric signature (1,4).[1] In metric signature (4,1), the set {γ0, γ1, γ2, γ3, γ5} is used, where the γμ are the appropriate ones for the (3,1) signature.[2] This pattern is repeated for spacetime dimension 2n even and the next odd dimension 2n + 1 for all n ≥ 1.[3] For more detail, see Higher-dimensional gamma matrices.
Identities
The following identities follow from the fundamental anticommutation relation, so they hold in any basis (although the last one depends on the sign choice for γ5{displaystyle gamma ^{5}}).
Miscellaneous identities
γμγμ=4I4{displaystyle gamma ^{mu }gamma _{mu }=4I_{4}}
Proof
Take the standard anticommutation relation:
- {γμ,γν}=γμγν+γνγμ=2ημνI4.{displaystyle left{gamma ^{mu },gamma ^{nu }right}=gamma ^{mu }gamma ^{nu }+gamma ^{nu }gamma ^{mu }=2eta ^{mu nu }I_{4}.}
One can make this situation look similar by using the metric η{displaystyle eta }:
- γμγμ=γμημνγν=ημνγμγν{displaystyle {begin{aligned}&gamma ^{mu }gamma _{mu }\={}&gamma ^{mu }eta _{mu nu }gamma ^{nu }=eta _{mu nu }gamma ^{mu }gamma ^{nu }end{aligned}}}
- =12(ημν+ηνμ)γμγν{displaystyle ={frac {1}{2}}left(eta _{mu nu }+eta _{nu mu }right)gamma ^{mu }gamma ^{nu }}
(η{displaystyle eta } symmetric)
- =12(ημνγμγν+ηνμγμγν){displaystyle ={frac {1}{2}}left(eta _{mu nu }gamma ^{mu }gamma ^{nu }+eta _{nu mu }gamma ^{mu }gamma ^{nu }right)}
(expanding)
- =12(ημνγμγν+ημνγνγμ){displaystyle ={frac {1}{2}}left(eta _{mu nu }gamma ^{mu }gamma ^{nu }+eta _{mu nu }gamma ^{nu }gamma ^{mu }right)}
(relabeling term on right)
- =12ημν{γμ,γν}{displaystyle ={frac {1}{2}}eta _{mu nu }left{gamma ^{mu },gamma ^{nu }right}}
- =12ημν(2ημνI4)=ημνημνI4=4I4.{displaystyle ={frac {1}{2}}eta _{mu nu }left(2eta ^{mu nu }I_{4}right)=eta _{mu nu }eta ^{mu nu }I_{4}=4I_{4}.}
γμγνγμ=−2γν{displaystyle gamma ^{mu }gamma ^{nu }gamma _{mu }=-2gamma ^{nu }}
Proof
Similarly to the proof of 1, again beginning with the standard commutation relation:
- γμγνγμ=γμ(2ημνI4−γμγν)=2γμημν−γμγμγν=2γν−4γν=−2γν.{displaystyle {begin{aligned}gamma ^{mu }gamma ^{nu }gamma _{mu }&=gamma ^{mu }left(2eta _{mu }^{nu }I_{4}-gamma _{mu }gamma ^{nu }right)\&=2gamma ^{mu }eta _{mu }^{nu }-gamma ^{mu }gamma _{mu }gamma ^{nu }\&=2gamma ^{nu }-4gamma ^{nu }=-2gamma ^{nu }.end{aligned}}}
γμγνγργμ=4ηνρI4{displaystyle gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma _{mu }=4eta ^{nu rho }I_{4}}
Proof
To show
- γμγνγργμ=4ηνρI4.{displaystyle gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma _{mu }=4eta ^{nu rho }I_{4}.}
Use the anticommutator to shift γμ{displaystyle gamma ^{mu }} to the right
γμγνγργμ{displaystyle gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma _{mu }}
={γμ,γν}γργμ−γνγμγργμ{displaystyle =left{gamma ^{mu },gamma ^{nu }right}gamma ^{rho }gamma _{mu }-gamma ^{nu }gamma ^{mu }gamma ^{rho }gamma _{mu }}
=2 ημνγργμ−γν{γμ,γρ}γμ+γνγργμγμ.{displaystyle =2 eta ^{mu nu }gamma ^{rho }gamma _{mu }-gamma ^{nu }left{gamma ^{mu },gamma ^{rho }right}gamma _{mu }+gamma ^{nu }gamma ^{rho }gamma ^{mu }gamma _{mu }.}
Using the relation γμγμ=4I{displaystyle gamma ^{mu }gamma _{mu }=4I} we can contract the last two gammas, and get
γμγνγργμ{displaystyle gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma _{mu }}
=2γργν−γν2ημργμ+4γνγρ{displaystyle =2gamma ^{rho }gamma ^{nu }-gamma ^{nu }2eta ^{mu rho }gamma _{mu }+4gamma ^{nu }gamma ^{rho }}
=2γργν−2γνγρ+4γνγρ{displaystyle =2gamma ^{rho }gamma ^{nu }-2gamma ^{nu }gamma ^{rho }+4gamma ^{nu }gamma ^{rho }}
=2(γργν+γνγρ){displaystyle =2left(gamma ^{rho }gamma ^{nu }+gamma ^{nu }gamma ^{rho }right)}
=2{γν,γρ}.{displaystyle =2left{gamma ^{nu },gamma ^{rho }right}.}
Finally using the anticommutator identity, we get
- γμγνγργμ=4ηνρI4.{displaystyle gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma _{mu }=4eta ^{nu rho }I_{4}.}
γμγνγργσγμ=−2γσγργν{displaystyle gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma }gamma _{mu }=-2gamma ^{sigma }gamma ^{rho }gamma ^{nu }}
Proof
γμγνγργσγμ{displaystyle gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma }gamma _{mu }}
=(2ημν−γνγμ)γργσγμ{displaystyle =(2eta ^{mu nu }-gamma ^{nu }gamma ^{mu })gamma ^{rho }gamma ^{sigma }gamma _{mu },} (anticommutator identity)
=2ημνγργσγμ−4γνηρσ{displaystyle =2eta ^{mu nu }gamma ^{rho }gamma ^{sigma }gamma _{mu }-4gamma ^{nu }eta ^{rho sigma },} (using identity 3)
=2γργσγν−4γνηρσ{displaystyle =2gamma ^{rho }gamma ^{sigma }gamma ^{nu }-4gamma ^{nu }eta ^{rho sigma }} (raising an index)
=2(2ηρσ−γσγρ)γν−4γνηρσ{displaystyle =2left(2eta ^{rho sigma }-gamma ^{sigma }gamma ^{rho }right)gamma ^{nu }-4gamma ^{nu }eta ^{rho sigma }} (anticommutator identity)
=−2γσγργν{displaystyle =-2gamma ^{sigma }gamma ^{rho }gamma ^{nu }} (2 terms cancel)
γμγνγρ=ημνγρ+ηνργμ−ημργν−iϵσμνργσγ5{displaystyle gamma ^{mu }gamma ^{nu }gamma ^{rho }=eta ^{mu nu }gamma ^{rho }+eta ^{nu rho }gamma ^{mu }-eta ^{mu rho }gamma ^{nu }-iepsilon ^{sigma mu nu rho }gamma _{sigma }gamma ^{5}}
Proof
If μ=ν=ρ{displaystyle mu =nu =rho } then ϵσμνρ=0{displaystyle epsilon ^{sigma mu nu rho }=0} and it is easy to verify the identity. That is the case also when μ=ν≠ρ{displaystyle mu =nu neq rho }, μ=ρ≠ν{displaystyle mu =rho neq nu } or ν=ρ≠μ{displaystyle nu =rho neq mu }.
On the other hand, if all three indices are different, ημν=0{displaystyle eta ^{mu nu }=0}, ημρ=0{displaystyle eta ^{mu rho }=0} and ηνρ=0{displaystyle eta ^{nu rho }=0} and both sides are completely antisymmetric; the left hand side because of the anticommutativity of the γ{displaystyle gamma } matrices, and on the right hand side because of the antisymmetry of ϵσμνρ{displaystyle epsilon _{sigma mu nu rho }}. It thus suffices to verify the identities for the cases of γ0γ1γ2{displaystyle gamma ^{0}gamma ^{1}gamma ^{2}}, γ0γ1γ3{displaystyle gamma ^{0}gamma ^{1}gamma ^{3}}, γ0γ2γ3{displaystyle gamma ^{0}gamma ^{2}gamma ^{3}} and γ1γ2γ3{displaystyle gamma ^{1}gamma ^{2}gamma ^{3}}.
- −iϵσ012γσγ5=−iϵ3012(−γ3)(iγ0γ1γ2γ3)=−ϵ3012γ0γ1γ2=ϵ0123γ0γ1γ2−iϵσ013γσγ5=−iϵ2013(−γ2)(iγ0γ1γ2γ3)=ϵ2013γ0γ1γ3=ϵ0123γ0γ1γ3−iϵσ023γσγ5=−iϵ1023(−γ1)(iγ0γ1γ2γ3)=−ϵ1023γ0γ2γ3=ϵ0123γ0γ2γ3−iϵσ123γσγ5=−iϵ0123(γ0)(iγ0γ1γ2γ3)=ϵ0123γ1γ2γ3{displaystyle {begin{aligned}-iepsilon ^{sigma 012}gamma _{sigma }gamma ^{5}&=-iepsilon ^{3012}left(-gamma ^{3}right)left(igamma ^{0}gamma ^{1}gamma ^{2}gamma ^{3}right)=-epsilon ^{3012}gamma ^{0}gamma ^{1}gamma ^{2}=epsilon ^{0123}gamma ^{0}gamma ^{1}gamma ^{2}\-iepsilon ^{sigma 013}gamma _{sigma }gamma ^{5}&=-iepsilon ^{2013}left(-gamma ^{2}right)left(igamma ^{0}gamma ^{1}gamma ^{2}gamma ^{3}right)=epsilon ^{2013}gamma ^{0}gamma ^{1}gamma ^{3}=epsilon ^{0123}gamma ^{0}gamma ^{1}gamma ^{3}\-iepsilon ^{sigma 023}gamma _{sigma }gamma ^{5}&=-iepsilon ^{1023}left(-gamma ^{1}right)left(igamma ^{0}gamma ^{1}gamma ^{2}gamma ^{3}right)=-epsilon ^{1023}gamma ^{0}gamma ^{2}gamma ^{3}=epsilon ^{0123}gamma ^{0}gamma ^{2}gamma ^{3}\-iepsilon ^{sigma 123}gamma _{sigma }gamma ^{5}&=-iepsilon ^{0123}left(gamma ^{0}right)left(igamma ^{0}gamma ^{1}gamma ^{2}gamma ^{3}right)=epsilon ^{0123}gamma ^{1}gamma ^{2}gamma ^{3}end{aligned}}}
Trace identities
The gamma matrices obey the following trace identities:
- tr(γμ)=0{displaystyle operatorname {tr} left(gamma ^{mu }right)=0}
- Trace of any product of an odd number of γμ{displaystyle gamma ^{mu }} is zero
- Trace of γ5{displaystyle gamma ^{5}} times a product of an odd number of γμ{displaystyle gamma ^{mu }} is still zero
- tr(γμγν)=4ημν{displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{nu }right)=4eta ^{mu nu }}
- tr(γμγνγργσ)=4(ημνηρσ−ημρηνσ+ημσηνρ){displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma }right)=4left(eta ^{mu nu }eta ^{rho sigma }-eta ^{mu rho }eta ^{nu sigma }+eta ^{mu sigma }eta ^{nu rho }right)}
- tr(γ5)=tr(γμγνγ5)=0{displaystyle operatorname {tr} left(gamma ^{5}right)=operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{5}right)=0}
- tr(γμγνγργσγ5)=−4iϵμνρσ{displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma }gamma ^{5}right)=-4iepsilon ^{mu nu rho sigma }}
- tr(γμ1…γμn)=tr(γμn…γμ1){displaystyle operatorname {tr} left(gamma ^{mu _{1}}dots gamma ^{mu _{n}}right)=operatorname {tr} left(gamma ^{mu _{n}}dots gamma ^{mu _{1}}right)}
Proving the above involves the use of three main properties of the trace operator:
- tr(A + B) = tr(A) + tr(B)
- tr(rA) = r tr(A)
- tr(ABC) = tr(CAB) = tr(BCA)
From the definition of the gamma matrices,
- γμγν+γνγμ=2ημνI{displaystyle gamma ^{mu }gamma ^{nu }+gamma ^{nu }gamma ^{mu }=2eta ^{mu nu }I}
We get
- γμγμ=ημμI{displaystyle gamma ^{mu }gamma ^{mu }=eta ^{mu mu }I}
or equivalently,
- γμγμημμ=I{displaystyle {frac {gamma ^{mu }gamma ^{mu }}{eta ^{mu mu }}}=I}
where ημμ{displaystyle eta ^{mu mu }} is a number, and γμγμ{displaystyle gamma ^{mu }gamma ^{mu }} is a matrix.
tr(γν)=1ημμtr(γνγμγμ){displaystyle operatorname {tr} (gamma ^{nu })={frac {1}{eta ^{mu mu }}}operatorname {tr} (gamma ^{nu }gamma ^{mu }gamma ^{mu })} (inserting the identity and using tr(rA) = r tr(A))
=−1ημμtr(γμγνγμ){displaystyle =-{frac {1}{eta ^{mu mu }}}operatorname {tr} (gamma ^{mu }gamma ^{nu }gamma ^{mu })} (from anti-commutation relation, and given that we are free to select μ≠ν{displaystyle mu neq nu })
=−1ημμtr(γνγμγμ){displaystyle =-{frac {1}{eta ^{mu mu }}}operatorname {tr} (gamma ^{nu }gamma ^{mu }gamma ^{mu })} (using tr(ABC) = tr(BCA))
=−tr(γν){displaystyle =-operatorname {tr} (gamma ^{nu })} (removing the identity)
This implies tr(γν)=0{displaystyle operatorname {tr} (gamma ^{nu })=0}
To show
- tr(odd num of γ)=0{displaystyle operatorname {tr} (mathrm {odd num of } gamma )=0}
First note that
- tr(γμ)=0.{displaystyle operatorname {tr} (gamma ^{mu })=0.}
We'll also use two facts about the fifth gamma matrix γ5{displaystyle gamma ^{5}} that says:
- (γ5)2=I4,andγμγ5=−γ5γμ{displaystyle left(gamma ^{5}right)^{2}=I_{4},quad mathrm {and} quad gamma ^{mu }gamma ^{5}=-gamma ^{5}gamma ^{mu }}
So lets use these two facts to prove this identity for the first non-trivial case: the trace of three gamma matrices. Step one is to put in one pair of γ5{displaystyle gamma ^{5}}'s in front of the three original γ{displaystyle gamma }'s, and step two is to swap the γ5{displaystyle gamma ^{5}} matrix back to the original position, after making use of the cyclicity of the trace.
tr(γμγνγρ){displaystyle operatorname {tr} (gamma ^{mu }gamma ^{nu }gamma ^{rho })}
=tr(γ5γ5γμγνγρ){displaystyle =operatorname {tr} left(gamma ^{5}gamma ^{5}gamma ^{mu }gamma ^{nu }gamma ^{rho }right)}
=−tr(γ5γμγνγργ5){displaystyle =-operatorname {tr} left(gamma ^{5}gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{5}right)}
=−tr(γ5γ5γμγνγρ){displaystyle =-operatorname {tr} left(gamma ^{5}gamma ^{5}gamma ^{mu }gamma ^{nu }gamma ^{rho }right)} (using tr(ABC) = tr(BCA))
=−tr(γμγνγρ){displaystyle =-operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{rho }right)}
This can only be fulfilled if
- tr(γμγνγρ)=0{displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{rho }right)=0}
The extension to 2n+1 (n integer) gamma matrices, is found by placing two gamma-5s after (say) the 2n-th gamma-matrix in the trace, commuting one out to the right (giving a minus sign) and commuting the other gamma-5 2n steps out to the left [with sign change (-1)^2n =1 ]. Then we use cyclic identity to get the two gamma-5s together, and hence they square to identity, leaving us with the trace equalling minus itself, i.e. 0.
If an odd number of gamma matrices appear in a trace followed by γ5{displaystyle gamma ^{5}}, our goal is to move γ5{displaystyle gamma ^{5}} from the right side to the left. This will leave the trace invariant by the cyclic property. In order to do this move, we must anticommute it with all of the other gamma matrices. This means that we anticommute it an odd number of times and pick up a minus sign. A trace equal to the negative of itself must be zero.
To show
- tr(γμγν)=4ημν{displaystyle operatorname {tr} (gamma ^{mu }gamma ^{nu })=4eta ^{mu nu }}
Begin with,
tr(γμγν){displaystyle operatorname {tr} (gamma ^{mu }gamma ^{nu })}
=12(tr(γμγν)+tr(γνγμ)){displaystyle ={frac {1}{2}}left(operatorname {tr} (gamma ^{mu }gamma ^{nu })+operatorname {tr} (gamma ^{nu }gamma ^{mu })right)}
=12tr(γμγν+γνγμ)=12tr({γμ,γν}){displaystyle ={frac {1}{2}}operatorname {tr} (gamma ^{mu }gamma ^{nu }+gamma ^{nu }gamma ^{mu })={frac {1}{2}}operatorname {tr} left(left{gamma ^{mu },gamma ^{nu }right}right)}
=122ημνtr(I4)=4ημν{displaystyle ={frac {1}{2}}2eta ^{mu nu }operatorname {tr} (I_{4})=4eta ^{mu nu }}
tr(γμγνγργσ){displaystyle operatorname {tr} (gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma })}
=tr(γμγν(2ηρσ−γσγρ)){displaystyle =operatorname {tr} left(gamma ^{mu }gamma ^{nu }(2eta ^{rho sigma }-gamma ^{sigma }gamma ^{rho })right)}
=2ηρσtr(γμγν)−tr(γμγνγσγρ)(1){displaystyle =2eta ^{rho sigma }operatorname {tr} left(gamma ^{mu }gamma ^{nu }right)-operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{sigma }gamma ^{rho }right)quad quad (1)}
For the term on the right, we'll continue the pattern of swapping γσ{displaystyle gamma ^{sigma }} with its neighbor to the left,
tr(γμγνγσγρ){displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{sigma }gamma ^{rho }right)}
=tr(γμ(2ηνσ−γσγν)γρ){displaystyle =operatorname {tr} left(gamma ^{mu }(2eta ^{nu sigma }-gamma ^{sigma }gamma ^{nu })gamma ^{rho }right)}
=2ηνσtr(γμγρ)−tr(γμγσγνγρ)(2){displaystyle =2eta ^{nu sigma }operatorname {tr} left(gamma ^{mu }gamma ^{rho }right)-operatorname {tr} left(gamma ^{mu }gamma ^{sigma }gamma ^{nu }gamma ^{rho }right)quad quad (2)}
Again, for the term on the right swap γσ{displaystyle gamma ^{sigma }} with its neighbor to the left,
tr(γμγσγνγρ){displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{sigma }gamma ^{nu }gamma ^{rho }right)}
=tr((2ημσ−γσγμ)γνγρ){displaystyle =operatorname {tr} left((2eta ^{mu sigma }-gamma ^{sigma }gamma ^{mu })gamma ^{nu }gamma ^{rho }right)}
=2ημσtr(γνγρ)−tr(γσγμγνγρ)(3){displaystyle =2eta ^{mu sigma }operatorname {tr} left(gamma ^{nu }gamma ^{rho }right)-operatorname {tr} left(gamma ^{sigma }gamma ^{mu }gamma ^{nu }gamma ^{rho }right)quad quad (3)}
Eq (3) is the term on the right of eq (2), and eq (2) is the term on the right of eq (1). We'll also use identity number 3 to simplify terms like so:
- 2ηρσtr(γμγν)=2ηρσ(4ημν)=8ηρσημν.{displaystyle 2eta ^{rho sigma }operatorname {tr} left(gamma ^{mu }gamma ^{nu }right)=2eta ^{rho sigma }(4eta ^{mu nu })=8eta ^{rho sigma }eta ^{mu nu }.}
So finally Eq (1), when you plug all this information in gives
tr(γμγνγργσ)=8ηρσημν−8ηνσημρ+8ημσηνρ{displaystyle operatorname {tr} (gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma })=8eta ^{rho sigma }eta ^{mu nu }-8eta ^{nu sigma }eta ^{mu rho }+8eta ^{mu sigma }eta ^{nu rho }}
- − tr(γσγμγνγρ)(4){displaystyle - operatorname {tr} left(gamma ^{sigma }gamma ^{mu }gamma ^{nu }gamma ^{rho }right)quad quad quad quad quad quad (4)}
The terms inside the trace can be cycled, so
- tr(γσγμγνγρ)=tr(γμγνγργσ).{displaystyle operatorname {tr} left(gamma ^{sigma }gamma ^{mu }gamma ^{nu }gamma ^{rho }right)=operatorname {tr} (gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma }).}
So really (4) is
- 2 tr(γμγνγργσ)=8ηρσημν−8ηνσημρ+8ημσηνρ{displaystyle 2 operatorname {tr} (gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma })=8eta ^{rho sigma }eta ^{mu nu }-8eta ^{nu sigma }eta ^{mu rho }+8eta ^{mu sigma }eta ^{nu rho }}
or
- tr(γμγνγργσ)=4(ηρσημν−ηνσημρ+ημσηνρ){displaystyle operatorname {tr} (gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma })=4left(eta ^{rho sigma }eta ^{mu nu }-eta ^{nu sigma }eta ^{mu rho }+eta ^{mu sigma }eta ^{nu rho }right)}
To show
tr(γ5)=0{displaystyle operatorname {tr} left(gamma ^{5}right)=0},
begin with
tr(γ5){displaystyle operatorname {tr} left(gamma ^{5}right)}
=tr(γ0γ0γ5){displaystyle =operatorname {tr} left(gamma ^{0}gamma ^{0}gamma ^{5}right)}
(because γ0γ0=I4{displaystyle gamma ^{0}gamma ^{0}=I_{4}})
=−tr(γ0γ5γ0){displaystyle =-operatorname {tr} left(gamma ^{0}gamma ^{5}gamma ^{0}right)}
(anti-commute the γ5{displaystyle gamma ^{5}} with γ0{displaystyle gamma ^{0}})
=−tr(γ0γ0γ5){displaystyle =-operatorname {tr} left(gamma ^{0}gamma ^{0}gamma ^{5}right)}
(rotate terms within trace)
=−tr(γ5){displaystyle =-operatorname {tr} left(gamma ^{5}right)}
(remove γ0{displaystyle gamma ^{0}}'s)
Add tr(γ5){displaystyle operatorname {tr} left(gamma ^{5}right)} to both sides of the above to see
2tr(γ5)=0{displaystyle 2operatorname {tr} left(gamma ^{5}right)=0}.
Now, this pattern can also be used to show
tr(γμγνγ5)=0{displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{5}right)=0}.
Simply add two factors of γα{displaystyle gamma ^{alpha }}, with α{displaystyle alpha } different from μ{displaystyle mu } and ν{displaystyle nu }. Anticommute three times instead of once, picking up three minus signs, and cycle using the cyclic property of the trace.
So,
tr(γμγνγ5)=0{displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{5}right)=0} .
For a proof of identity 6, the same trick still works unless (μνρσ){displaystyle left(mu nu rho sigma right)} is some permutation of (0123), so that all 4 gammas appear. The anticommutation rules imply that interchanging two of the indices changes the sign of the trace, so tr(γμγνγργσγ5){displaystyle operatorname {tr} left(gamma ^{mu }gamma ^{nu }gamma ^{rho }gamma ^{sigma }gamma ^{5}right)} must be proportional to ϵμνρσ{displaystyle epsilon ^{mu nu rho sigma }} (ϵ0123=η0μη1νη2ρη3σϵμνρσ=η00η11η22η33ϵ0123=−1){displaystyle left(epsilon ^{0123}=eta ^{0mu }eta ^{1nu }eta ^{2rho }eta ^{3sigma }epsilon _{mu nu rho sigma }=eta ^{00}eta ^{11}eta ^{22}eta ^{33}epsilon _{0123}=-1right)}. The proportionality constant is 4i{displaystyle 4i}, as can be checked by plugging in (μνρσ)=(0123){displaystyle (mu nu rho sigma )=(0123)}, writing out γ5{displaystyle gamma ^{5}}, and remembering that the trace of the identity is 4.
Denote the product of n{displaystyle n} gamma matrices by Γ=γμ1γμ2…γμn.{displaystyle Gamma =gamma ^{mu 1}gamma ^{mu 2}dots gamma ^{mu n}.} Consider the Hermitian conjugate of Γ{displaystyle Gamma }:
Γ†{displaystyle Gamma ^{dagger }}
=γμn†…γμ2†γμ1†{displaystyle =gamma ^{mu ndagger }dots gamma ^{mu 2dagger }gamma ^{mu 1dagger }}
=γ0γμnγ0…γ0γμ2γ0γ0γμ1γ0{displaystyle =gamma ^{0}gamma ^{mu n}gamma ^{0}dots gamma ^{0}gamma ^{mu 2}gamma ^{0}gamma ^{0}gamma ^{mu 1}gamma ^{0}}
(since conjugating a gamma matrix with γ0{displaystyle gamma ^{0}} produces its Hermitian conjugate as described below)
=γ0γμn…γμ2γμ1γ0{displaystyle =gamma ^{0}gamma ^{mu n}dots gamma ^{mu 2}gamma ^{mu 1}gamma ^{0}}
(all γ0{displaystyle gamma ^{0}}s except the first and the last drop out)
Conjugating with γ0{displaystyle gamma ^{0}} one more time to get rid of the two γ0{displaystyle gamma ^{0}}s that are there, we see that γ0Γ†γ0{displaystyle gamma ^{0}Gamma ^{dagger }gamma ^{0}} is the reverse of Γ{displaystyle Gamma }. Now,
tr(γ0Γ†γ0){displaystyle operatorname {tr} left(gamma ^{0}Gamma ^{dagger }gamma ^{0}right)}
=tr(Γ†){displaystyle =operatorname {tr} left(Gamma ^{dagger }right)}
(since trace is invariant under similarity transformations)
=tr(Γ∗){displaystyle =operatorname {tr} left(Gamma ^{*}right)}
(since trace is invariant under transposition)
=tr(Γ){displaystyle =operatorname {tr} left(Gamma right)}
(since the trace of a product of gamma matrices is real)
Normalization
The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however. We can impose
(γ0)†=γ0{displaystyle left(gamma ^{0}right)^{dagger }=gamma ^{0}}, compatible with (γ0)2=I4{displaystyle left(gamma ^{0}right)^{2}=I_{4}}
and for the other gamma matrices (for k = 1, 2, 3)
(γk)†=−γk{displaystyle left(gamma ^{k}right)^{dagger }=-gamma ^{k}}, compatible with (γk)2=−I4.{displaystyle left(gamma ^{k}right)^{2}=-I_{4}.}
One checks immediately that these hermiticity relations hold for the Dirac representation.
The above conditions can be combined in the relation
- (γμ)†=γ0γμγ0.{displaystyle left(gamma ^{mu }right)^{dagger }=gamma ^{0}gamma ^{mu }gamma ^{0}.}
The hermiticity conditions are not invariant under the action γμ→S(Λ)γμS(Λ)−1{displaystyle gamma ^{mu }to S(Lambda )gamma ^{mu }{S(Lambda )}^{-1}} of a Lorentz transformation Λ{displaystyle Lambda } because S(Λ){displaystyle S(Lambda )} is not necessarily a unitary transformation due to the non-compactness of the Lorentz group.
Feynman slash notation used in Quantum field theory
The Feynman slash notation is defined by
- a/:=γμaμ{displaystyle {a!!!/}:=gamma ^{mu }a_{mu }}
for any 4-vector a.
Here are some similar identities to the ones above, but involving slash notation:
- a/b/=a⋅b−iaμσμνbν{displaystyle {a!!!/}{b!!!/}=acdot b-ia_{mu }sigma ^{mu nu }b_{nu }}
- a/a/=aμaνγμγν=12aμaν(γμγν+γνγμ)=ημνaμaν=a2{displaystyle {a!!!/}{a!!!/}=a^{mu }a^{nu }gamma _{mu }gamma _{nu }={frac {1}{2}}a^{mu }a^{nu }left(gamma _{mu }gamma _{nu }+gamma _{nu }gamma _{mu }right)=eta _{mu nu }a^{mu }a^{nu }=a^{2}}
- tr(a/b/)=4(a⋅b){displaystyle operatorname {tr} left({a!!!/}{b!!!/}right)=4(acdot b)}
- tr(a/b/c/d/)=4[(a⋅b)(c⋅d)−(a⋅c)(b⋅d)+(a⋅d)(b⋅c)]{displaystyle operatorname {tr} left({a!!!/}{b!!!/}{c!!!/}{d!!!/}right)=4left[(acdot b)(ccdot d)-(acdot c)(bcdot d)+(acdot d)(bcdot c)right]}
- tr(γ5a/b/)=0{displaystyle operatorname {tr} left(gamma _{5}{a!!!/}{b!!!/}right)=0}
- tr(γ5a/b/c/d/)=−4iϵμνρσaμbνcρdσ{displaystyle operatorname {tr} left(gamma _{5}{a!!!/}{b!!!/}{c!!!/}{d!!!/}right)=-4iepsilon _{mu nu rho sigma }a^{mu }b^{nu }c^{rho }d^{sigma }}
- γμa/γμ=−2a/{displaystyle gamma _{mu }{a!!!/}gamma ^{mu }=-2{a!!!/}}
- γμa/b/γμ=4a⋅b{displaystyle gamma _{mu }{a!!!/}{b!!!/}gamma ^{mu }=4acdot b}
γμa/b/c/γμ=−2c/b/a/{displaystyle gamma _{mu }{a!!!/}{b!!!/}{c!!!/}gamma ^{mu }=-2{c!!!/}{b!!!/}{a!!!/}}
- where ϵμνρσ{displaystyle epsilon _{mu nu rho sigma }} is the Levi-Civita symbol and σμν=i2[γμ,γν].{displaystyle sigma ^{mu nu }={frac {i}{2}}left[gamma ^{mu },gamma ^{nu }right].} Actually traces of products of odd number of γ{displaystyle gamma } is zero and thus
tr(a/)=tr(a/b/c/)=tr(a/b/c/d/e/)=0{displaystyle operatorname {tr} left({a!!!/}right)=operatorname {tr} left({a!!!/}{b!!!/}{c!!!/}right)=operatorname {tr} left({a!!!/}{b!!!/}{c!!!/}{d!!!/}{e!!!/}right)=0}[4]
Other representations
The matrices are also sometimes written using the 2×2 identity matrix, I2{displaystyle I_{2}}, and
- γk=(0σk−σk0){displaystyle gamma ^{k}={begin{pmatrix}0&sigma ^{k}\-sigma ^{k}&0end{pmatrix}}}
where k runs from 1 to 3 and the σk are Pauli matrices.
Dirac basis
The gamma matrices we have written so far are appropriate for acting on Dirac spinors written in the Dirac basis; in fact, the Dirac basis is defined by these matrices. To summarize, in the Dirac basis:
- γ0=(I200−I2),γk=(0σk−σk0),γ5=(0I2I20).{displaystyle gamma ^{0}={begin{pmatrix}I_{2}&0\0&-I_{2}end{pmatrix}},quad gamma ^{k}={begin{pmatrix}0&sigma ^{k}\-sigma ^{k}&0end{pmatrix}},quad gamma ^{5}={begin{pmatrix}0&I_{2}\I_{2}&0end{pmatrix}}.}
Weyl (chiral) basis
Another common choice is the Weyl or chiral basis,[5] in which γk{displaystyle gamma ^{k}} remains the same but γ0{displaystyle gamma ^{0}} is different, and so γ5{displaystyle gamma ^{5}} is also different, and diagonal,
- γ0=(0I2I20),γk=(0σk−σk0),γ5=(−I200I2),{displaystyle gamma ^{0}={begin{pmatrix}0&I_{2}\I_{2}&0end{pmatrix}},quad gamma ^{k}={begin{pmatrix}0&sigma ^{k}\-sigma ^{k}&0end{pmatrix}},quad gamma ^{5}={begin{pmatrix}-I_{2}&0\0&I_{2}end{pmatrix}},}
or in more compact notation:
- γμ=(0σμσ¯μ0),σμ≡(1,σi),σ¯μ≡(1,−σi).{displaystyle gamma ^{mu }={begin{pmatrix}0&sigma ^{mu }\{bar {sigma }}^{mu }&0end{pmatrix}},quad sigma ^{mu }equiv (1,sigma ^{i}),quad {bar {sigma }}^{mu }equiv (1,-sigma ^{i}).}
The Weyl basis has the advantage that its chiral projections take a simple form,
- ψL=12(1−γ5)ψ=(I2000)ψ,ψR=12(1+γ5)ψ=(000I2)ψ.{displaystyle psi _{L}={frac {1}{2}}left(1-gamma ^{5}right)psi ={begin{pmatrix}I_{2}&0\0&0end{pmatrix}}psi ,quad psi _{R}={frac {1}{2}}left(1+gamma ^{5}right)psi ={begin{pmatrix}0&0\0&I_{2}end{pmatrix}}psi .}
The idempotence of the chiral projections is manifest.
By slightly abusing the notation and reusing the symbols ψL/R{displaystyle psi _{L/R}} we can then identify
- ψ=(ψLψR),{displaystyle psi ={begin{pmatrix}psi _{L}\psi _{R}end{pmatrix}},}
where now ψL{displaystyle psi _{L}} and ψR{displaystyle psi _{R}} are left-handed and right-handed two-component Weyl spinors.
Another possible choice[6] of the Weyl basis has
- γ0=(0−I2−I20),γk=(0σk−σk0),γ5=(I200−I2).{displaystyle gamma ^{0}={begin{pmatrix}0&-I_{2}\-I_{2}&0end{pmatrix}},quad gamma ^{k}={begin{pmatrix}0&sigma ^{k}\-sigma ^{k}&0end{pmatrix}},quad gamma ^{5}={begin{pmatrix}I_{2}&0\0&-I_{2}end{pmatrix}}.}
The chiral projections take a slightly different form from the other Weyl choice,
- ψR=(I2000)ψ,ψL=(000I2)ψ.{displaystyle psi _{R}={begin{pmatrix}I_{2}&0\0&0end{pmatrix}}psi ,quad psi _{L}={begin{pmatrix}0&0\0&I_{2}end{pmatrix}}psi .}
In other words,
- ψ=(ψRψL),{displaystyle psi ={begin{pmatrix}psi _{R}\psi _{L}end{pmatrix}},}
where ψL{displaystyle psi _{L}} and ψR{displaystyle psi _{R}} are the left-handed and right-handed two-component Weyl spinors, as before.
Majorana basis
There is also the Majorana basis, in which all of the Dirac matrices are imaginary, and the spinors and Dirac equation are real. Regarding the Pauli matrices, the basis can be written as
- γ0=(0σ2σ20),γ1=(iσ300iσ3),γ2=(0−σ2σ20),γ3=(−iσ100−iσ1),γ5=(σ200−σ2),C=(0−iσ2−iσ20).{displaystyle {begin{aligned}gamma ^{0}&={begin{pmatrix}0&sigma ^{2}\sigma ^{2}&0end{pmatrix}},&gamma ^{1}&={begin{pmatrix}isigma ^{3}&0\0&isigma ^{3}end{pmatrix}},&gamma ^{2}&={begin{pmatrix}0&-sigma ^{2}\sigma ^{2}&0end{pmatrix}},\gamma ^{3}&={begin{pmatrix}-isigma ^{1}&0\0&-isigma ^{1}end{pmatrix}},&gamma ^{5}&={begin{pmatrix}sigma ^{2}&0\0&-sigma ^{2}end{pmatrix}},&C&={begin{pmatrix}0&-isigma ^{2}\-isigma ^{2}&0end{pmatrix}}.end{aligned}}}
The reason for making all gamma matrices imaginary is solely to obtain the particle physics metric (+, −, −, −) in which squared masses are positive. The Majorana representation, however, is real. One can factor out the i{displaystyle i} to obtain a different representation with four component real spinors and real gamma matrices. The consequence of removing the i{displaystyle i} is that the only possible metric with real gamma matrices is (−, +, +, +).
Cℓ1,3(C) and Cℓ1,3(R)
The Dirac algebra can be regarded as a complexification of the real algebra Cℓ1,3(R), called the space time algebra:
- Cl1,3(C)=Cl1,3(R)⊗C{displaystyle Cl_{1,3}(mathbb {C} )=Cl_{1,3}(mathbb {R} )otimes mathbb {C} }
Cℓ1,3(R) differs from Cℓ1,3(C): in Cℓ1,3(R) only real linear combinations of the gamma matrices and their products are allowed.
Two things deserve to be pointed out. As Clifford algebras, Cℓ1,3(C) and Cℓ4(C) are isomorphic, see classification of Clifford algebras. The reason is that the underlying signature of the spacetime metric loses its signature (3,1) upon passing to the complexification. However, the transformation required to bring the bilinear form to the complex canonical form is not a Lorentz transformation and hence not "permissible" (at the very least impractical) since all physics is tightly knit to the Lorentz symmetry and it is preferable to keep it manifest.
Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible (and usually enlightening) to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in a real Clifford algebra that square to −1, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.[7]
However, in contemporary practice, the Dirac algebra rather than the space-time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.
Euclidean Dirac matrices
In quantum field theory one can Wick rotate the time axis to transit from Minkowski space to Euclidean space. This is particularly useful in some renormalization procedures as well as lattice gauge theory. In Euclidean space, there are two commonly used representations of Dirac matrices:
Chiral representation
- γ1,2,3=(0iσ1,2,3−iσ1,2,30),γ4=(0I2I20){displaystyle gamma ^{1,2,3}={begin{pmatrix}0&isigma ^{1,2,3}\-isigma ^{1,2,3}&0end{pmatrix}},quad gamma ^{4}={begin{pmatrix}0&I_{2}\I_{2}&0end{pmatrix}}}
Notice that the factors of i{displaystyle i} have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra
- {γμ,γν}=2δμνI4{displaystyle left{gamma ^{mu },gamma ^{nu }right}=2delta ^{mu nu }I_{4}}
will emerge. It is also worth noting that there are variants of this which insert instead −i{displaystyle -i} on one of the matrices, such as in lattice QCD codes which use the chiral basis.
In Euclidean space,
- γM5=i(γ0γ1γ2γ3)M=1i2(γ4γ1γ2γ3)E=(γ1γ2γ3γ4)E=γE5.{displaystyle gamma _{M}^{5}=ileft(gamma ^{0}gamma ^{1}gamma ^{2}gamma ^{3}right)_{M}={frac {1}{i^{2}}}left(gamma ^{4}gamma ^{1}gamma ^{2}gamma ^{3}right)_{E}=left(gamma ^{1}gamma ^{2}gamma ^{3}gamma ^{4}right)_{E}=gamma _{E}^{5}.}
Using the anti-commutator and noting that in Euclidean space (γμ)†=γμ{displaystyle left(gamma ^{mu }right)^{dagger }=gamma ^{mu }}, one shows that
- (γ5)†=γ5{displaystyle left(gamma ^{5}right)^{dagger }=gamma ^{5}}
In chiral basis in Euclidean space,
- γ5=(−I200I2){displaystyle gamma ^{5}={begin{pmatrix}-I_{2}&0\0&I_{2}end{pmatrix}}}
which is unchanged from its Minkowski version.
Non-relativistic representation
- γ1,2,3=(0−iσ1,2,3iσ1,2,30),γ4=(I200−I2),γ5=(0−I2−I20){displaystyle gamma ^{1,2,3}={begin{pmatrix}0&-isigma ^{1,2,3}\isigma ^{1,2,3}&0end{pmatrix}},quad gamma ^{4}={begin{pmatrix}I_{2}&0\0&-I_{2}end{pmatrix}},quad gamma ^{5}={begin{pmatrix}0&-I_{2}\-I_{2}&0end{pmatrix}}}
See also
- Pauli matrices
- Gell-Mann matrices
- Higher-dimensional gamma matrices
- Fierz identity
References
^ The reason for the notation γ5 is because that set of matrices (ΓA) = (γμ, iγ5) with A = (0, 1, 2, 3, 4) satisfy the five-dimensional Clifford algebra {ΓA, ΓB} = 2ηAB. Tong 2007, p. 93.
^ Weinberg 2002 Section 5.5.
^ de Wit & Smith 1996, p. 679.
^ Lecture note from University of Texas at Austin
^ The matrices in this basis, provided below, are the similarity transforms of the Dirac basis matrices of the previous paragraph, U†γDμU{displaystyle U^{dagger }gamma _{D}^{mu }U}, where U=12(1−γ5γ0)=12(II−II){displaystyle U={frac {1}{sqrt {2}}}left(1-gamma ^{5}gamma ^{0}right)={frac {1}{sqrt {2}}}{begin{pmatrix}I&I\-I&Iend{pmatrix}}}.
^ Michio Kaku, Quantum Field Theory, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
ISBN 0-19-509158-2, appendix A
^ See e.g. Hestenes 1996.
Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.
- A. Zee, Quantum Field Theory in a Nutshell (2003), Princeton University Press: Princeton, New Jersey.
ISBN 0-691-01019-6. See chapter II.1. - M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995)
ISBN 0-201-50397-2 See chapter 3.2.
W. Pauli (1936). "Contributions mathématiques à la théorie des matrices de Dirac". Annales de l'Institut Henri Poincaré. 6: 109.
Weinberg, S. (2002), The Quantum Theory of Fields, 1, Cambridge University Press, ISBN 0-521-55001-7
Tong, David (2007). "Quantum Field Theory". David Tong at University of Cambridge. p. 93. Retrieved 2015-03-07.
de Wit, B.; Smith, J. (1986). Field Theory in Particle Physics. North-Holland Personal Library. 1. North-Holland. ISBN 978-0444869999.
Appendix E
David Hestenes, Real Dirac Theory, in J. Keller and Z. Oziewicz (Eds.), The Theory of the Electron, UNAM, Facultad de Estudios Superiores, Cuautitlan, Mexico (1996), pp. 1–50.
External links
Dirac matrices on mathworld including their group properties- Dirac matrices as an abstract group on GroupNames
Hazewinkel, Michiel, ed. (2001) [1994], "Dirac matrices", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4