Moore2_corrected

Appendix C Lorentz group and the Dirac algebra This appendix provides a review and summary of the Lorentz gr oup, its properties, and the properties of its infinitesimal generat ors. It then reviews representations of the Lorentz group and the Dirac algebra. This material is intended to supplement Chapter 1, for those students who are not as familiar with the Lorentz group and Dirac equation as they find they nee d to be. C.1 Lorentz group According to special relativity, physical laws are unchang ed by a linear change of coordinates, x′µ= Λµνxν+ξµ(C.1) with Λ and ξreal, provided it leave unchanged the invariant separation between points, (x−y)µ(x−y)µ=ηµν(x−y)µ(x−y)ν=−[(x−y)0]2+ [/vector x−/vector y]2. This condition does not constrain ξ, since it cancels in the difference, but it imposes a constraint on Λ, xµxµ=x′ µx′µ=ηµνΛµαxαΛνβxβ, (C.2) for allxµ. A transformation of the form shown in Eq. (C.1) which satisfi es Eq. (C.2) is called a Poincar´ e transformation. These trans formations close and form a group, called the Poincar´ e group. The subgroup wh ere Λ is the identity matrix and ξis arbitrary is a subgroup called the group of transla- tions. We assume that this group and its implications, such a s conservation of energy and momentum, are familiar to the reader. Instead w e concentrate on the subgroup in which ξ= 0, which is called the Lorentz group. It is convenient to think of an element of the Lorentz group as a matrix 494 C.1 Lorentz group 495 which is operating on the coordinate xµ. This is possible if we always write Λ with its first index raised and second index lowered, so it ca rriesxµto x′µ, both with raised index. Writing in this way, repeated Loren tz trans- formations are implemented via matrix multiplications of t he respective Λ matrices: x′µ= Λµνxνandx′′µ= Λ′µνx′ν⇒x′′µ= Λ′µνΛναxα≡/bracketleftig Λ′Λ/bracketrightigµ αxα. (C.3) We see from Eq. (C.2) that the condition on Λµνto be a Lorentz trans- formation is, ηµνxµxν=ηαβΛαµΛβνxµxν(C.4) for allxµ. Since this must hold for all xµ, we have ηµν= ΛαµηαβΛβν, (C.5) or (writing ηµνasηwhen using matrix notation) η= ΛTηΛ. (C.6) The group of matrices Λ satisfying Eq. (C.6) is called O(3,1), and is a Lie group. Therefore the same technology of Lie algebra generat ion may be applied to it as to the groups of the previous appendix. As we will discuss momentarily, not all elements of O(3,1) can be built infinitesimally from the identity. Those elements which can , form a subgroup writtenSO(3,1), which we will now analyze. A Lorentz transformation Λµν which is infinitesimally close to the identity must be of form , Λµν=δµ ν+ωµν, (C.7) withωµνa matrix of infinitesimal coefficients. The condition on ωµνfor Λµνto be a valid Lorentz transformation is found by inserting Eq . (C.7) into Eq. (C.6) and expanding to linear order in ω: ηµν=/parenleftig δα µ+ωαµ/parenrightig ηαβ/parenleftig δβ ν+ωβν/parenrightig =ηµν+ (ωνµ+ωµν) +O(ω2), 0 =ωνµ+ωµν. (C.8) That is, the condition on ωµνis thatωµνbe antisymmetric on its indices. The space of real antisymmetric 4 ×4 matrices is 6 dimensional, so the Lorentz group is 6 dimensional. Nowωµνis related to ωµνasωµν=ηµαωαν. Sinceη00=−1 andηii= 1 fori= 1,2,3, the sign of the space-time component of ωµνmust be the same 496 Lorentz group and the Dirac algebra as the sign of the time-space component, while the space-spa ce components must be antisymmetric. Thus, the most general form of ωµνis, ωµν= 0b1b2b3 b10−r3r2 b2r30−r1 b3−r2r10 , (C.9) symmetric in the space-time entries and anti-symmetric in t he space-space entries. The b1,b2,b3entries respectively cause infinitesimal boosts in the 1,2,3 directions; the r1,r2,r3entries cause infinitesimal rotations about the 1,2,3 axes. A general element of SO(3,1) can be written as an exponential of afiniteωµν, Λµν= (expω)µ ν=δµν+ωµν+1 2ωµαωαν+1 6ωµαωαβωβν+... . (C.10) When only the riare nonzero, this gives a rotation by angle |/vector r|about the ˆraxis. When only the biare nonzero, this gives a boost by velocity tanh |/vectorb| along the ˆbaxis. When both /vector rand/vectorbare nonzero the Lorentz transformation cannot be described either solely as a rotation or as a boost. Note that, while a rotation by angle |/vector r|= 2πgives the identity Λ, no nonzero magnitude of boost|/vectorb|returns the identity. Hence the group SO(3,1) isnon-compact . Now we argue that O(3,1) has 4 disconnected pieces, one of which is SO(3,1). To see this, take the determinant of Eq. (C.6): Detη= Det ΛTηΛ = Detη×(Det Λ)2. (C.11) Sinceηis nonsingular, we can divide by Det η: (Det Λ)2= 1 ⇒Det Λ = ±1. (C.12) The determinant must vary continuously within a path connec ted region of O(3,1), but you cannot go discontinuously from 1 to −1, so any elements ofO(3,1) with Det Λ = −1 cannot be elements of the connected group SO(3,1). An element of O(3,1) with Det Λ = 1 is called proper , and an element with Det Λ = −1 is called improper . Furthermore, if we write out the µ= 0,ν= 0 element of Eq. (C.6), it is η00= Λµ0ηµνΛν0 −1 = −Λ00Λ00+/summationdisplay i=1,2,3Λi0Λi0 (Λ00)2= 1 +/summationdisplay i=1,2,3(Λi0)2≥1, (C.13) so the square of the time-time component of any Λ must always b e at least C.2 Generators of the Lorentz group 497 1, and Λ00must be either ≥1 or≤ −1. Again, you cannot go continuously from≥1 to≤ −1, so no elements of SO(3,1) have Λ00<0. An element of O(3,1) with Λ00≥1 is called orthochronous , and an element with Λ00≤ −1 is called non-orthochronous . The canonical example of an improper (but orthochronous) el ement of O(3,1) is the parity transformation , P= 1 0 0 0 0−1 0 0 0 0 −1 0 0 0 0 −1 , (C.14) which satisfies Eq. (C.6) but has determinant −1. The canonical example of a non-orthochronous (and also improper) transformation is thetime reversal transformation , T= −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , (C.15) which also satisfies Eq. (C.6) but has T00=−1. It turns out that any element of O(3,1) must be an element of SO(3,1), times either the iden- tity (proper orthochronous), P(improper orthochronous), T(improper non- orthochronous), or PT(proper non-orthochronous). The improper or non- orthochronous Lorentz transformations need not be symmetr ies of nature–in fact, in the standard model, they are not–but it is an axiom of field theory that the elements of SO(3,1) must be symmetries. C.2 Generators of the Lorentz group As discussed in Section B.1, each element Λ ∈SO(3,1) must have associated with it a unitary operator U(Λ) which implements it on the Hilbert space, and which represents the group operation, U(Λ1)U(Λ2) =U(Λ1Λ2). (C.16) For an element infinitesimally close to the identity, it must be possible to expand these operators in a Lie algebra of generators, U(ω) =1+i 2ωµνˆJµν+O(ω2), (C.17) 498 Lorentz group and the Dirac algebra for some operators ˆJµν, antisymmetric in µ,ν. Similarly, there are genera- tors for translations, U(ξ) =1−iξµˆPµ. (C.18) TheˆPiare also called momentum operators, and the ˆJijare called angular momentum operators. The commutation relations between the operators ˆPµ,ˆJµνcan be worked out by using Eq. (C.16). For instance, consider a translatio n by a small dis- tanceξµ, either preceded or followed by a Lorentz transformation in volving ωνα. We will evaluate the difference between the two orders of ope ration, in two ways. First, if we translate first and then rotate, then the coordin ate is trans- formed according to x′µ=xµ+ξµ, x′′µ= (δµν+ωµν)(xν+ξν) =xµ+ (ωµνxν) + (ξµ+ωµνξν), (C.19) where the first and second parenthesis represent a rotation a nd a translation. The result is the same rotation, and a translation by ξµplus an extra piece involvingωandξ. If the rotation is performed first, we get x′µ=xµ+ωµνxν, x′′µ=xµ+ (ωµνxν) + (ξµ), (C.20) which is the rotation and the translation just by ξ. The unitary operators for these transformations are, U(ωξ) = 1+i 2ωµνˆJµν−iηµν(ξµ+ωµαξα)ˆPν, U(ξω) = 1+i 2ωµνˆJµν−iηµν(ξµ)ˆPν. (C.21) The difference of the operators, to second order in the infinit esimals, is U(ωξ)−U(ξω) =−iηµνωµαξαˆPν. (C.22) (There is actually also a second order in ωpiece, but it is the same for the twoU’s and therefore cancels in this difference.) Alternately, we can say using Eq. (C.16) that U(ωξ) =U(ω)U(ξ) =/parenleftbigg 1+i 2ωµνˆJµν/parenrightbigg/parenleftbigg 1−iηαβξαˆPβ/parenrightbigg , U(ξω) =U(ξ)U(ω) =/parenleftbigg 1−iηαβξαˆPβ/parenrightbigg/parenleftbigg 1+i 2ωµνˆJµν/parenrightbigg , C.2 Generators of the Lorentz group 499 U(ωξ)−U(ξω) = −i2 2ωµνηαβξα/bracketleftigˆJµν,ˆPβ/bracketrightig . (C.23) Now equating Eq. (C.22) and Eq. (C.23), we learn what the comm utator of ˆPwith ˆJmust be: −i2 2ωµνξα/bracketleftigˆJµν,ˆPα/bracketrightig =−iωµνξαηναˆPµ. (C.24) This must hold for any antisymmetric ωµνand anyξα, so (antisymmetrizing over the indices on ω) the operators must satisfy /bracketleftigˆJµν,ˆPα/bracketrightig =i/parenleftig ηµαˆPν−ηναˆPµ/parenrightig . (C.25) By a completely analogous but more involved procedure one ca n also show, /bracketleftigˆJµν,ˆJαβ/bracketrightig =i/parenleftig ηνβˆJµα+ηµαˆJνβ−ηµβˆJνα−ηναˆJµβ/parenrightig , (C.26) and (this is simpler) /bracketleftigˆPµ,ˆPν/bracketrightig = 0. (C.27) These commutation relations are called the Poincar´ e algeb ra. To make contact with the more familiar generators of rotatio ns and boosts, it is convenient to define ˆJi≡ǫijk 2ˆJjk, (C.28) ˆKi≡ˆJ0i, (C.29) which are respectively the generator of rotations about the iaxis and of boosts along the iaxis, so a rotation by /vectorθis exp( −iJiθi) and a boost by /vector vis exp( −iKivi). They satisfy the commutation relations, following from Eq. (C.26), /bracketleftigˆJi,ˆJj/bracketrightig =iǫijkˆJk, (C.30) /bracketleftigˆJi,ˆKj/bracketrightig =iǫijkˆKk, (C.31) /bracketleftigˆKi,ˆKj/bracketrightig =−iǫijkˆJk. (C.32) The first expression is the familiar commutator between rota tions. The second means that, if a rotation is performed before a boost, the boost will be in a different direction than before the rotation is perfor med, which is intuitively clear. The third result is more surprising; the commutator of two boosts is a rotation. More importantly, the sign is opposite on the last result than on the previous two. 500 Lorentz group and the Dirac algebra C.3 Representations of the Lorentz group Just as for an internal symmetry, an SO(3,1) transformation will carry a field to a linear combination of fields, so the fields must trans form under representations of the group. The difference is that the tran sformed field will be at the Lorentz transformed point: U(ω)ϕa(x)U∗(ω) =D−1 ab(ω)ϕb(Λx), (C.33) withD−1(ω) =D(−ω) anωdependent matrix in the space of fields. The fields can be chosen to block-diagonalize the matrix Dinto irreducible rep- resentations of the Lorentz group. For instance, in QED, the components of the gauge potential Aµmix with each other under Lorentz transformations, but they never mix with the different spin components of the el ectronei; so there is one “block” of Dwhich mixes the Aµand an independent block mixing the ei. Our goal now is to find the possible structures Dcan take. Just as for internal symmetries, there are two very simple ir reducible rep- resentations, which are also physically important. One is t he trivial (singlet) representation, U(ω)φ(x)U∗(ω) =φ(Λx), (C.34) which forSO(3,1) is called the scalar representation . Lorentz symmetry demands that the Lagrangian be a Lorentz scalar. The other is thevec- tor representation , for which the field index is a four-vector index and the representation matrix is Λ itself: U(ω)Aµ(x)U∗(ω) = (Λ−1)µνAν(Λx). (C.35) A representation is determined by a set of matrices Jµνwith the same commutation relations as the ˆJµν. That is, the matrix Dabmust be of the form Dab(ω) = exp/parenleftbiggi 2ωµνJµν ab/parenrightbigg , (C.36) with the exponentiation interpreted as matrix exponentiat ion witha,bthe matrix indices, and Jµνsatisfying /bracketleftig Jµν,Jαβ/bracketrightig =i/parenleftig ηνβJµα+ηµαJνβ−ηµβJνα−ηναJµβ/parenrightig .(C.37) The problem of finding representations is the problem of findi ng all sets of matrices with this algebra. It is believed that only field theories containing finite numb ers of fields are well defined. Therefore we need only look for finite-dimensio nal representa- tions ofSO(3,1). The classification of the representations is made easier by C.3 Representations of the Lorentz group 501 the following convenient property of the group. Using Eq. (C .30)–Eq. (C.32), one can show that the operators ˆLi≡ˆJi+iˆKi 2, ˆRi≡ˆJi−iˆKi 2, (C.38) satisfy the commutation relations, /bracketleftigˆLi,ˆLj/bracketrightig =iǫijkˆLk, (C.39) /bracketleftigˆRi,ˆRj/bracketrightig =iǫijkˆRk, (C.40) /bracketleftigˆLi,ˆRj/bracketrightig = 0. (C.41) Therefore the generators of SO(3,1) can be split into two subsets which commute with each other, and each satisfy the same commutati on relations as the group SU(2). This group is familiar as the group of rotations and its representations are well known; they are the spin-zero repr esentation, the spin-half representation, the spin-one representation, a nd so forth. A general irreducible representation can be described by its transfo rmation properties under ˆLand under ˆR,eg, spinm/2 under ˆLand spinn/2 under ˆR. Only four representations will be of any interest to us in stu dying the standard model, because it turns out that only four represen tations can participate in renormalizable interactions in a theory sat isfying the basic principles laid out in Section 1.2. The first of these is the scalar representation already intro duced. The scalar representation transforms as (0 ,0), that is, as spin-zero under ˆLand spin-zero under ˆR. The Lie algebra representations are ˆJµν= 0 and the transformation matrix D= 1 is the identity. The second common representation is the vector representat ion, which transforms as (1 2,1 2). The Lie algebra is represented as Jµν αβ=−i(ηµαην β− ηµ βηνα), andD= Λ, as already discussed. Note that for both of these representations, the matrix Dis always real; therefore it is consistent to consider real valued scalar or vector fields. The other two interesting representations are called spinor representa- tions, and consist of two fields which mix with each other under Lore ntz transformations. Since these are probably less familiar to the reader and are in some ways more complicated than the scalar and vector r epresenta- tions, we will discuss them at length in the next section. 502 Lorentz group and the Dirac algebra C.4 Spinors and the Dirac algebra We now introduce the other two physically important represe ntations of the Lorentz group, the left and right handed spinor representat ions. A field transforming in one of these can be rewritten in terms of the o ther, and it is convenient to combine them together using Majorana notation , which we will also introduce and which we use throughout this book. C.4.1 Spinor representations The simplest nontrivial matrices which satisfy Eq. (C.39) a re the Pauli ma- trices, σ1=/parenleftigg 0 1 1 0/parenrightigg , σ 2=/parenleftigg 0−i i0/parenrightigg , σ 3=/parenleftigg 1 0 0−1/parenrightigg ,(C.42) which satisfy the commutation relation /bracketleftigσi 2,σj 2/bracketrightig =iǫijkσk 2. (C.43) Therefore, if the matrices representing ˆLiandˆRiareσi/2 and 0 respec- tively, we get a representation of the Lorentz algebra. Inve rting Eq. (C.38), rotations and boosts are implemented by the matrices, Ji=σi 2,Ki=−iσi 2,(Left handed spinor) (C.44) which it is easy to show satisfy Eq. (C.30) through Eq. (C.32) . Therefore, a pair of fields ψa,a= 1,2 can transform under Lorentz trans- formations according to U(−ω)ψaU∗(−ω) =Dab(ω)ψb, D(ω) = [exp( −i(ri−ibi)σi/2)],(C.45) whereri,biare the amount of rotation and boost performed, as introduce d in Eq. (C.9). The two fields ψaare generally referred to as the components of a single spinor field withathespinor index , which is almost always suppressed by writing ψandDin matrix notation ( ψas a column vector, Das a matrix). Such a spinor field is called a left handed Weyl spinor ψL. Alternately, Ricould be represented by the Pauli matrices and Liby 0’s, Ji=σi 2,Ki=iσi 2,(Right handed spinor) (C.46) in which case a Lorentz transform acts on ψvia U(−ω)ψaU∗(−ω) =Dab(ω)ψb, D(ω) = [exp( −i(ri+ibi)σi/2)].(C.47) A field transforming this way is called a right handed Weyl spinor ψR. C.4 Spinors and the Dirac algebra 503 Since the matrices Dwe just constructed are in general complex, a spinor ψLorψRmust be a pair of complex fields. We can ask how the complex conjugate of ψLtransforms. Because complex conjugation flips the iin front ofKin Eq. (C.44), the answer is that it transforms as a right hand ed Weyl spinor. More properly, defining the matrix ǫ≡iσ2=/parenleftigg 0 1 −1 0/parenrightigg satisfying ǫσ∗ i=−σiǫ, (C.48) we see that ǫtimes the conjugate of ψLtransforms according to, U(−ω)ǫψ∗ LU∗(−ω) =ǫ/parenleftbigg exp/bracketleftbigg −i(ri−ibi)σi 2/bracketrightbigg ψL/parenrightbigg∗ =ǫexp/bracketleftbigg +i(ri+ibi)σ∗ i 2/bracketrightbigg ψ∗ L = exp/bracketleftbigg −i(ri+ibi)σi 2/bracketrightbigg ǫψ∗ L, (C.49) which is precisely the transformation rule for a right hande d Weyl spinor. Similarly, −ǫψ∗ Rtransforms as a left handed Weyl spinor, and −ǫ(ǫψ∗ L)∗=ψL transforms as a left handed Weyl spinor again. Both the field a nd its complex conjugate will typically appear in the Lagrangian so it is im portant to have a notation which can deal with each. Whether we consider the l eft or right handed version as the field rather than the conjugated object is a matter of convention. C.4.2 Weyl, Majorana, Dirac There are two common notational ways of dealing with the fact that a field can be written either as a left or a right handed spinor. One, called Weyl notation , expresses the fields as two component objects, and then specifies whether one is referring to ψLor to its right handed conjugateǫψ∗ Lby using either an undotted or a dotted index: ψα=ψLand ψ˙α=ǫψ∗ L. (Indices are raised and lowered using ǫand dotted and undotted according to whether they are conjugated.) This notation is common in the supersymmetry and string theory literature. An alternative which we will use, Majorana notation , writes a single four component field ψM, defined as ψM=/parenleftigg ψL ǫψ∗ L/parenrightigg , (C.50) that is,ψMredundantly records both the left handed and the right hande d 504 Lorentz group and the Dirac algebra ways of writing the field. The individual pieces can be access ed separately by using the projection operators PL≡/parenleftigg 10 0 0/parenrightigg andPR≡/parenleftigg 0 0 01/parenrightigg . (C.51) The action of rotations and boosts on ψMare respectively, Ji=/parenleftiggσi 20 0σi 2/parenrightigg ,Ki=/parenleftigg−iσi 20 0iσi 2/parenrightigg . (C.52) If a left-handed spinor transforms nontrivially under an in ternal symmetry group, then since the right-handed version involves comple x conjugation, the right-handed version ǫψ∗ Ltransforms under the conjugate representation. In particular, if ψLhas charge qunder aU(1) symmetry and is in the fundamental representation of an SU(N) symmetry, then ǫψ∗ Lhas charge −qand transforms under the anti-fundamental representation ofSU(N). One must keep this in mind when constructing Lagrangians out of Majorana spinors. In QED and QCD, if we write the spinor fields as left-handed obj ects, the fields form pairs with conjugate symmetry transformation pr operties. For instance, in QED, there is a field ELwhich is charge -1 under Uem(1), called the left-handed electron, and a field −ǫE∗ Rwhich is charge 1 under Uem(1), called the left-handed positron. In this case it is most conv enient to think of the latter as the conjugate of a right-handed field with cha rge -1,ER, called the right-handed electron, and to combine them toget her in a single 4-component object called a Dirac spinor ,e= [ELER]T. The Lorentz transformation properties of Majorana and Dira c spinors are the same. The two distinctions are that the upper and lower co mponents of a Dirac spinor generally have the same transformation pro perties under internal symmetries, while for Majorana spinors they have c onjugate trans- formation properties; and the upper and lower components of a Dirac spinor are independent, while for a Majorana spinor they are redund ant notations for the same field. C.4.3 Tensor products of spinors Since the Lagrangian must be a Lorentz scalar, it must be a sum of terms even in spinorial fields. Therefore we need to know how produc ts of two spinor fields transform. We will only consider the combinati on of a spinor fieldψ1with the complex conjugate of another, ψ† 2. This is sufficient for Majorana spinors because ψT 2ψ1can be re-expressed in terms of ψ† 2ψ1, and C.4 Spinors and the Dirac algebra 505 it suffices for Dirac spinors with internal symmetries becaus e only such com- binations are invariant under the internal symmetries. The Hermitian conjugate of a spinor field ψtransforms as U(−ω)ψ†U∗(−ω) = (D(ω)ψ)†=ψ†D†(ω). (C.53) TheJiare Hermitian, but the Kiare anti-Hermitian, so D(ω) is not in general unitary. Therefore ψ†does not have the inverse transformation property of ψ. However, there is a Hermitian, unit determinant matrix β, β≡/parenleftigg 01 10/parenrightigg , β Ji=Jiβ, β Ki=−Kiβ, (C.54) which flips the sign of Kbut not Jwhen commuted across D†, soD†β= βD−1. Therefore, defining ψ=ψ†β, called the Dirac conjugate ofψ, U(−ω)ψU∗(−ω) =ψ†D†(ω)β=ψ†βD−1(ω) =ψD−1(ω), (C.55) soψhas the inverse transformation property of ψ. Sinceψhas 4 components, there are sixteen independent 4 ×4 matrices Γ which can be used to combine spinors, ψ2Γψ1. These can all be gotten from four such matrices, called the gamma matrices γµ, given in Eq. (1.87). These satisfy anticommutation relations called the Clifford algebra , /braceleftig γµ, γν/bracerightig = 2ηµν1. (C.56) The matrices Jµνcan be expressed in terms of the gamma matrices: Jµν=−i 4/bracketleftig γµ, γν/bracketrightig , (C.57) which together with Eq. (C.56) is enough to prove that Jµνsatisfies the Lorentz algebra, Eq. (C.37). Further, these relations ensu re that /bracketleftig Jµν, γα/bracketrightig =i(ηµαγν−ηναγµ), (C.58) from which it follows that D−1(ω)γµD(ω) = Λµνγν. (C.59) Therefore, while the combination ψ2ψ1is a scalar, U(−ω)ψ2ψ1U−1(−ω) =ψ2D−1(ω)D(ω)ψ=ψ2ψ1is a scalar,(C.60) the combination ψ2γµψ1is a vector, U(−ω)ψ2γµψ1U−1(−ω) =ψ2D−1(ω)γµD(ω)ψ1= Λµνψ2γνψ1is a vector. (C.61) 506 Lorentz group and the Dirac algebra Similarly, defining σµν= 2iJµν, the combination U(−ω)ψ2σµνψ1U−1(−ω) = ΛµαΛνβψ2σαβψ1 (C.62) is a rank-2 antisymmetric tensor. Next, define γ5=γ5≡i 24ǫµναβγµγνγαγβ=−iγ0γ1γ2γ3, (C.63) where the latter follows from the anti-commutation of the di stinct gamma matrices. We have that U(−ω)ψ2γ5ψ1U−1(−ω) =i 24ǫµναβψ2D−1γµγνγαγβDψ1 (C.64) =i 24ǫµναβΛµσΛνρΛακΛβζψ2γσγργκγζψ1 = (Det Λ)i 24ǫσρκζψ2γσγργκγζψ1 = (Det Λ) ψ2γ5ψ1. Thereforeψ2γ5ψ1is apseudoscalar , a scalar under SO(3,1) which flips sign under parity transformations. Finally, the quantity ψ2γµγ5ψtransforms as a pseudovector, U(−ω)ψ2γµγ5ψ1U−1(−ω) =ψ2D−1γµγ5Dψ1= (Det Λ)Λµνψ2γνγ5ψ1. (C.65) Since this gives 1 + 4 + 6 + 4 + 1 = 16 independent contractions, t he above are exhaustive; any other matrix sandwiched between ψ2andψ1must be a linear combination of 1,γµ,σµν,γµγ5, andγ5. The choice of matrices made above is called the chiral basis a nd is con- venient because the right and left handed components of ψfactorize. How- ever, multiplying ψby an arbitrary unitary matrix Sand all matrices by Γ→SΓS−1leaves the theory unchanged. While the explicit expression s for the matrices are obviously changed, certain relations are n ot, and are there- fore particularly valuable. In particular, the Clifford alg ebra, Eq. (C.56), the relations Eq. (C.57), Eq. (C.58), Eq. (C.59), the definit ion Eq. (C.63) of γ5in terms of the other γmatrices, and the relations between the projection operators and γ5, PL=1+γ5 2, P R=1−γ5 2, (C.66) are basis independent and should therefore be sufficient to ev aluate any invariant quantities.