Gamma matrices

In mathematical physics, the gamma matrices, , 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

is the time-like, hermitian matrix. The other three are space-like, antihermitian matrices. More compactly, , and , where denotes the Kronecker product and the (for j = 1, 2, 3) denote the Pauli matrices.

The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, 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.

Mathematical structureEdit

The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation

 

where   is the anticommutator,   is the Minkowski metric with signature (+ − − −), and   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

 

and Einstein notation is assumed.

Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation:

 

or a multiplication of all gamma matrices by  , 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

 

Physical structureEdit

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), 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) 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) 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) ≈ End(Ux). If S belongs to a representation of the Lorentz group, then the induced action ESES−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) given by

 

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. The last identity can be recognized as the defining relationship for matrices belonging to an indefinite orthogonal group, which is   written in indexed notation. This means that quantities of the form

 

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

 

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 = γμγνγνγμ 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. The spin representation of the Lorentz group is encoded in the spin group Spin(1, 3) (for real, uncharged spinors) and in the complexified spin group Spinc(1, 3) for charged (Dirac) spinors.

Expressing the Dirac equationEdit

In natural units, the Dirac equation may be written as

 

where   is a Dirac spinor.

Switching to Feynman notation, the Dirac equation is

 

The fifth "gamma" matrix, γ5Edit

It is useful to define a product of the four gamma matrices as  , so that

  (in the Dirac basis).

Although   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   was called " ".

  has also an alternative form:

 

using the convention  , or

 

using the convention  . Proof:

This can be seen by exploiting the fact that all the four gamma matrices anticommute, so

 ,

where   is the type (4,4) generalized Kronecker delta in 4 dimensions, in full antisymmetrization. If   denotes the Levi-Civita symbol in n dimensions, we can use the identity  . Then we get, using the convention  ,

 

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:

 .

Some properties are:

  • It is hermitian:
     
  • Its eigenvalues are ±1, because:
     
  • It anticommutes with the four gamma matrices:
     

In fact,   and   are eigenvectors of   since

 , and  

Five dimensionsEdit

The Clifford algebra in odd dimensions behaves like two copies of the Clifford algebra of one less dimension, a left copy and a right copy.[1] Thus, one can employ a bit of a trick to repurpose iγ5 as one of the generators of the Clifford algebra in five dimensions. In this case, the set {γ0, γ1, γ2, γ3, 5} 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).[2] 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.[3] This pattern is repeated for spacetime dimension 2n even and the next odd dimension 2n + 1 for all n ≥ 1.[4] For more detail, see Higher-dimensional gamma matrices.

IdentitiesEdit

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  ).

Miscellaneous identitiesEdit

  1.  

    Proof:

    Take the standard anticommutation relation:

     

    One can make this situation look similar by using the metric  :

     
     
      (  symmetric)
      (expanding)
      (relabeling term on right)
     
     
  2.  

    Proof:

    Similarly to the proof of 1, again beginning with the standard commutation relation:

     
  3.  

    Proof:

    To show

     

    Use the anticommutator to shift   to the right

     
     
     

    Using the relation   we can contract the last two gammas, and get

     
     
     
     
     

    Finally using the anticommutator identity, we get

     
  4.  

    Proof:

     
      (anticommutator identity)
      (using identity 3)
      (raising an index)
      (anticommutator identity)
      (2 terms cancel)
  5.  

    Proof:

    If   then   and it is easy to verify the identity. That is the case also when  ,   or  .

    On the other hand, if all three indices are different,  ,   and   and both sides are completely antisymmetric; the left hand side because of the anticommutativity of the   matrices, and on the right hand side because of the antisymmetry of  . It thus suffices to verify the identities for the cases of  ,  ,   and  .

     
  6.  

Trace identitiesEdit

The gamma matrices obey the following trace identities:

  1.  
  2. Trace of any product of an odd number of   is zero
  3. Trace of   times a product of an odd number of   is still zero
  4.  
  5.  
  6.  
  7.  
  8.  

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)

Proof of 0:

From the definition of the gamma matrices,

 

We get

 

or equivalently,

 

where   is a number, and   is a matrix.

  (inserting the identity and using tr(rA) = r tr(A))
  (from anti-commutation relation, and given that we are free to select  )
  (using tr(ABC) = tr(BCA))
  (removing the identity)

This implies  

Proof of 1:

To show

 

First note that

 

We'll also use two facts about the fifth gamma matrix   that says:

 

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  's in front of the three original  's, and step two is to swap the   matrix back to the original position, after making use of the cyclicity of the trace.

   
 
  (using tr(ABC) = tr(BCA))
 

This can only be fulfilled if

 

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.

Proof of 2:

If an odd number of gamma matrices appear in a trace followed by  , our goal is to move   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.

Proof of 3:

To show

 

Begin with,

   
 
 

Proof of 4:

   
 

For the term on the right, we'll continue the pattern of swapping   with its neighbor to the left,

   
 

Again, for the term on the right swap   with its neighbor to the left,

   
 

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:

 

So finally Eq (1), when you plug all this information in gives

 
 

The terms inside the trace can be cycled, so

 

So really (4) is

 

or

 

Proof of 5:

To show

 ,

begin with

    (because  )
  (anti-commute the   with  )
  (rotate terms within trace)
  (remove  's)

Add   to both sides of the above to see

 .

Now, this pattern can also be used to show

 .

Simply add two factors of  , with   different from   and  . Anticommute three times instead of once, picking up three minus signs, and cycle using the cyclic property of the trace.

So,

 .

Proof of 6:

For a proof of identity 6, the same trick still works unless   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   must be proportional to    . The proportionality constant is  , as can be checked by plugging in  , writing out  , and remembering that the trace of the identity is 4.

Proof of 7:

Denote the product of   gamma matrices by   Consider the Hermitian conjugate of  :

   
  (since conjugating a gamma matrix with   produces its Hermitian conjugate as described below)
  (all  s except the first and the last drop out)

Conjugating with   one more time to get rid of the two  s that are there, we see that   is the reverse of  . Now,

    (since trace is invariant under similarity transformations)
  (since trace is invariant under transposition)
  (since the trace of a product of gamma matrices is real)

NormalizationEdit

The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however. We can impose

 , compatible with  

and for the other gamma matrices (for k = 1, 2, 3)

 , compatible with  

One checks immediately that these hermiticity relations hold for the Dirac representation.

The above conditions can be combined in the relation

 

The hermiticity conditions are not invariant under the action   of a Lorentz transformation   because   is not necessarily a unitary transformation due to the non-compactness of the Lorentz group.

Charge conjugationEdit

The charge conjugation operator, in any basis, may be defined as

 

where   denotes the matrix transpose. The explicit form that   takes is dependent on the specific representation chosen for the gamma matrices (its form expressed as product of the gamma matrices is representation independent, but the gamma matrices themselves have different forms in different representation). This is because although charge conjugation is an automorphism of the gamma group, it is not an inner automorphism (of the group). Conjugating matrices can be found, but they are representation-dependent.

Representation-independent identities include:

 

In addition, for all four representations given below (Dirac, Majorana and both chiral variants), one has

 

Feynman slash notation used in quantum field theoryEdit

The Feynman slash notation is defined by

 

for any 4-vector a.

Here are some similar identities to the ones above, but involving slash notation:

  •  
  •  
  •  
  •  
  •  
  •  
  •  
  •  
  •  
    where   is the Levi-Civita symbol and   Actually traces of products of odd number of   is zero and thus
  •  [5]

Other representationsEdit

The matrices are also sometimes written using the 2×2 identity matrix,  , and

 

where k runs from 1 to 3 and the σk are Pauli matrices.

Dirac basisEdit

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:

 

In the Dirac basis, the charge conjugation operator is[6]

 

Weyl (chiral) basisEdit

Another common choice is the Weyl or chiral basis, in which   remains the same but   is different, and so   is also different, and diagonal,

 

or in more compact notation:

 

The Weyl basis has the advantage that its chiral projections take a simple form,

 

The idempotence of the chiral projections is manifest.

By slightly abusing the notation and reusing the symbols   we can then identify

 

where now   and   are left-handed and right-handed two-component Weyl spinors.

The charge conjugation operator in this basis is

 

The Dirac basis can be obtained from the Weyl basis as

 

via the unitary transform

 

Weyl (chiral) basis (alternate form)Edit

Another possible choice[6][7] of the Weyl basis has

 

The chiral projections take a slightly different form from the other Weyl choice,

 

In other words,

 

where   and   are the left-handed and right-handed two-component Weyl spinors, as before.

The charge conjugation operator in this basis is

 

This basis can be obtained from the Dirac basis above as   via the unitary transform

 

Majorana basisEdit

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[6]

 

where   is the charge conjugation matrix, as defined above.

(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 to obtain a different representation with four component real spinors and real gamma matrices. The consequence of removing the   is that the only possible metric with real gamma matrices is (−, +, +, +).)

The Majorana basis can be obtained from the Dirac basis above as   via the unitary transform

 

Cl1,3(C) and Cl1,3(R)Edit

The Dirac algebra can be regarded as a complexification of the real algebra Cl1,3(R), called the space time algebra:

 

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 (1,3) 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.[8]

In the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Clp,q() for arbitrary dimensions p,q; the anti-commutation of the Weyl spinors emerges naturally from the Clifford algebra.[9] The Weyl spinors transform under the action of the spin group  . The complexification of the spin group, called the spinc group  , is a product   of the spin group with the circle   The product   just a notational device to identify   with   The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the   component, which can be identified with the   fiber of the electromagnetic interaction. The   is entangling parity and charge conjugation in a manner suitable for relating the Dirac particle/anti-particle states (equivalently, the chiral states in the Weyl basis). The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field. This is in contrast to the Majorana spinor and the ELKO spinor, which cannot (i.e. they are electrically neutral), as they explicitly constrain the spinor so as to not interact with the   part coming from the complexification.

Insofar as the presentation of charge and parity can be a confusing topic in conventional quantum field theory textbooks, the more careful dissection of these topics in a general geometric setting can be elucidating. Standard expositions of the Clifford algebra construct the Weyl spinors from first principles; that they "automatically" anti-commute is an elegant geometric by-product of the construction, completely by-passing any arguments that appeal to the Pauli exclusion principle (or the sometimes common sensation that Grassmann variables have been introduced via ad hoc argumentation.)

However, in contemporary practice in physics, 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 matricesEdit

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 representationEdit

 

Notice that the factors of   have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra

 

will emerge. It is also worth noting that there are variants of this which insert instead   on one of the matrices, such as in lattice QCD codes which use the chiral basis.

In Euclidean space,

 

Using the anti-commutator and noting that in Euclidean space  , one shows that

 

In chiral basis in Euclidean space,

 

which is unchanged from its Minkowski version.

Non-relativistic representationEdit

 

See alsoEdit

ReferencesEdit

  1. ^ Jurgen Jost (2002) "Riemannian Geometry and Geometric Analysis (3rd Edition)" Springer Universitext (See Corollary 1.8.1, page 68)
  2. ^ The set of matrices (ΓA) = (γμ, 5) with A = (0, 1, 2, 3, 4) satisfy the five-dimensional Clifford algebra {ΓA, ΓB} = 2ηAB. See Tong 2007, p. 93.
  3. ^ Weinberg 2002 Section 5.5.
  4. ^ de Wit & Smith 1996, p. 679.
  5. ^ Lecture note from University of Texas at Austin
  6. ^ a b c Claude Itzykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory", MacGraw-Hill (See Appendix A)
  7. ^ Michio Kaku, Quantum Field Theory, ISBN 0-19-509158-2, appendix A
  8. ^ See e.g. Hestenes 1996.
  9. ^ Jurgen Jost (2002) "Riemannian Geometry and Geometric Analysis (3rd Edition)", Springer Universitext. See section 1.8

External linksEdit