Spinor Formulation of Relativistic Quantum Mechanics

Chapter Spinor Formulation of Relativistic Quantum Mechanics. The Lorentz Transformation of the Dirac Bispinor We will provide in the following a new formulation of the Dirac equation in the chiral representation defined through 0.5 0.9. Starting point is the Lorentz transformation S w ϑ for the bispinor wave function Ψ in the chiral representation as given by 0.6. This transformation can be written a z 0 S z. 0 b z a z exp z σ. b z exp z σ.3 z w i ϑ..4 We have altered here slightly our notation of S w ϑ expressing its dependence on w ϑ through a complex variable z z C 3. Because of its block-diagonal form each of the diagonal components of S z i.e. a z and b z must be two-dimensional irreducible representations of the Lorentz group. This fact is remarkable since it implies that the representations provided through a z and b z are of lower dimension then the four-dimensional natural representation L w ϑ µ ν. The lower dimensionality of a z and b z implies in a sense that the corresponding representation of the Lorentz group is more basic than the natural representation and may serve as a building block for all representations in particular may be exploited to express the Lorentz-invariant equations of relativistic quantum mechanics. This is indeed what will be achieved in the following. We will proceed by building as much as possible on the results obtained sofar in the chiral representation of the Dirac equation. We will characterize the space on which the transformations a z We will see below that the representations a z and b z are in fact isomorphic to the natural representation i.e. different L w ϑ µ ν correspond to different a z and b z. 355

356 Spinor Formulation and b z act the so-called spinor space will establish the map between L w ϑ µ ν and a z b z express 4-vectors A µ A ν the operator µ and the Pauli matrices σ in the new representation and finally formulate the Dirac equation neutrino equation and the Klein Gordon equation in the spinor representation. A First Characterization of the Bispinor States We note that in case w 0 the Dirac transformations are pure rotations. In this case a z and b z are identical and read ai ϑ bi ϑ exp ϑ σ θ R 3..5 The transformations in this case i.e. for z iϑ ϑ R 3 are elements of SU and correspond in fact to the rotational transformations of spin- states as described by D m m ϑ usually expressed as product of rotations and of functions of Euler angles α β γ see Chapter 5. For ϑ 0 β 0 T the transformations are ai β ê bi β ê d mm β cos β sin β sin β cos β..6 as given by 5.43. This characterization allows one to draw conclusions regarding the state space in which a z and b z operate namely a space of vectors state state for which holds state + state z iϑ ϑ R 3.7 where stands for transforms like. Here ± represents the familiar spin states. Since a z acts on the first two components of the solution Ψ of the Dirac equation and since b z acts on the third and fourth component of Ψ one can characterize Ψ Ψ x µ + Ψ x µ Ψ 3 x µ + Ψ 4 x µ z i ϑ ϑ R 3..8 We like to stress that there exists however a distinct difference in the transformation behaviour of Ψ x µ Ψ x µ and Ψ 3 x µ Ψ 4 x µ in case z w + i ϑ for w 0. In this case holds a z b z and Ψ x µ Ψ x µ transform according to a z whereas Ψ 3 x µ Ψ 4 x µ transform according to b z. Relationship Between a z and b z The transformation b z can be related to the conjugate complex of the transformation a z i.e. to a z exp z σ.9

.: Lorentz Transformation of Dirac Bispinor 357 where σ σ σ σ 3. One can readily verify [c.f. 5.4] From this one can derive where σ σ σ σ σ 3 σ 3..0 ɛ 0 0 b z ɛ a z ɛ. ɛ 0 0.. To prove. one first demonstrates that for ɛ and ɛ as given in. does in fact hold ɛ ɛ. One notices then using ɛfaɛ fɛaɛ [ ɛ a zɛ exp z ɛ σ ɛ ]..3 Explicit matrix multiplication using 5.4.0. yields or in short Similarly one can show ɛσ ɛ σ σ ɛσ ɛ σ σ.4 ɛσ 3 ɛ σ 3 σ 3 ɛ σɛ σ..5 ɛ σɛ σ.6 a result to be used further below. Hence one can express ɛ a z ɛ exp z σ b z..7 We conclude therefore that the transformation. can be written in the form a z 0 S z 0 ɛ a z ɛ.8 with a z given by..4 and ɛ ɛ given by.. This demonstrates that a z is the transformation which characterizes both components of S z. Spatial Inversion One may question from the form of S z why the Dirac equation needs to be four-dimensional featuring the components Ψ x µ Ψ x µ as well as Ψ 3 x µ Ψ4 x µ even though these pairs of components transform independently of each other. The answer lies in the necessity that application of spatial inversion should transform a solution of the Dirac equation into another possible solution of the Dirac equation. The effect of inversion on Lorentz transformations is however that they alter w into w but leave rotation angles ϑ unaltered.

358 Spinor Formulation Let P denote the representation of spatial inversion in the space of the wave functions Ψ. Obviously P i.e. P P. The transformation S z in the transformed space is then P S w + i ϑ P S w + i ϑ b z 0 0 a z i.e. the transformations a z and b z become interchanged. This implies P Ψ Ψ Ψ 3 Ψ 4 Ψ 3 Ψ 4 Ψ Ψ.9..0 Obviously the space spanned by only two of the components of Ψ is not invariant under spatial inversion and hence does not suffice for particles like the electron which obey inversion symmetry. However for particles like the neutrinos which do not obey inversion symmetry two components of the wave function are sufficient. In fact the Lorentz invariant equation for neutrinos is only dimensional.. Relationship Between the Lie Groups SLC and SO3 We have pointed out that ai ϑ ϑ R 3 which describes pure rotations is an element of SU. However a w + i ϑ for w 0 is an element of SL C { M M is a complex matrix detm }.. One can verify this by evaluating the determinant of a z det a z det e z σ e tr z σ. which follows from the fact that for any complex non-singular matrix M holds det e M e trm.3 and from [c.f. 5.4] tr σ j 0 j 3..4 Exercise..: Show that SL C defined in. together with matrix multiplication as the binary operation forms a group. The proof of this important property is straightforward in case of hermitian M see Chapter 5. For the general case the proof based on the Jordan Chevalley theorem can be found in G.G.A. Bäuerle and E.A. de Kerf Lie Algebras Part Elsevier Amsterdam 990 Exercise.0.3.

.: Lie Groups SLC and SO3 359 Mapping A µ onto matrices MA µ We want to establish now the relationship between SL C and the group L + of proper orthochronous Lorentz transformations. Starting point is a bijective map between R 4 and the set of two-dimensional hermitian matrices defined through where MA µ σ µ A µ.5 0 0 0 i 0 σ µ 0 0 i 0 0..6 }{{}}{{}}{{}}{{} σ 0 σ σ σ 3 The quantity σ µ thus defined does not transform like a covariant 4 vector. In fact one wishes that the definition.5 of the matrix MA µ is independent of the frame of reference i.e. in a transformed frame should hold σ µ A µ L ρ ν σ µ A µ..7 Straightforward transformation into another frame of reference would replace σ µ by σ µ. Using A µ L µ νa ν one would expect in a transformed frame to hold σ µ A µ Lρ ν σ µ L µ ν A ν..8 Consistency of.8 and.7 requires then L ν µ σ µ σ ν.9 where we used 0.76. Noting that for covariant vectors according to 0.75 holds a ν L µ ν a µ one realizes that σ µ transforms inversely to covariant 4 vectors. We will prove below [cf..35] this transformation behaviour. MA µ according to.5 can also be written MA µ A 0 + A 3 A ia A + ia A 0 A 3..30 Since the components of A µ are real the matrix MA µ is hermitian as can be seen from inspection of.6 or from the fact that the matrices σ 0 σ σ σ 3 are hermitian. The function MA µ is bijective in fact one can provide a simple expression for the inverse of MA µ M MA µ A µ tr M σ µ..3 Exercise..: Show that σ 0 σ σ σ 3 provide a linear independent basis for the space of hermitian matrices. Argue why MA µ σ µ A µ provides a bijective map. Demonstrate that.3 holds. The following important property holds for MA µ which follows directly from.30. det MA µ A µ A µ.3

360 Spinor Formulation Transforming the matrices MA µ We define now a transformation of the matrix MA µ in the space of hermitian matrices M a M a M a a SL C..33 This transformation conserves the hermitian property of M since M a M a a M a a M a M.34 where we used the properties M M and a a. Due to deta the transformation.33 conserves the determinant of M. In fact it holds for the matrix M defined through.33 detm det a M a deta deta detm [deta] detm detm..35 We now apply the transformation.33 to MA µ describing the action of the transformation in terms of transformations of A µ. In fact for any a SL C and for any A µ there exists an A µ such that MA µ a MA µ a..36 The suitable A µ can readily be constructed using.3. Accordingly any a SL C defines the transformation [c.f..3] A µ a A µ tr a MA µ a σ µ..37 Because of.3.35 holds for this transformation A µ A µ A µ A µ.38 which implies that.37 defines actually a Lorentz transformation. The linear character of the transformation becomes apparent expressing A µ as given in.37 using.5 A µ tr a σ ν a σ µ A ν.39 which allows us to express finally A µ La µ ν A ν ; La µ ν tr a σ ν a σ µ..40 Exercise..3: Show that La µ ν defined in.40 is an element of L +.

.: Lie Groups SLC and SO3 36 La µ ν provides a homomorphism We want to demonstrate now that the map between SL C and SO3 defined through La µ ν [cf..40] respects the group property of SL C and of SO3 i.e. L µ ρ La µ ν La µ ρ }{{} product in SO3 L a a }{{ } µ ρ.4 product in SLC For this purpose one writes using trab trba La µ ν La µ ρ ν ν tr a σ ν a σ µ tr a σ ρ a σ ν tr σ ν a σ µa tr a σ ρ a σ ν..4 Defining Γ a σ µa Γ a σ ρ a.43 and using the definition of L µ ρ in.4 results in L µ ρ 4 σ ν αβ Γ βα Γ γδ σ ν δγ A αβγδ Γ βα Γ γδ.44 4 ναβ γδ αβ γδ where A αβγδ ν σ ν αβ σ ν δγ..45 One can demonstrate through direct evaluation A αβγδ α β γ δ α β γ δ α γ β δ α γ β δ 0 else.46 which yields L µ ρ Γ Γ + Γ Γ + Γ Γ + Γ Γ tr ΓΓ tr a σ µa a σ ρ a tr σ µ a a σ ρ a a tr a a σ ρ a a σ µ La a µ ρ..47 This completes the proof of the homomorphic property of La µ ν.

36 Spinor Formulation Generators for SL C which correspond to K J The transformations a SL C as complex matrices are defined through four complex or correspondingly eight real numbers. Because of the condition deta only six independent real numbers actually suffice for the definition of a. One expects then that six generators G j and six real coordinates f j can be defined which allow one to represent a in the form 6 a exp f j G j..48 j We want to determine now the generators of the transformation a z as defined in. which correspond to the generators K K K 3 J J J 3 of the Lorentz transformations L µ ν in the natural representations i.e. correspond to the generators given by 0.47 0.48. To this end we consider infinitesimal transformations and keep only terms of zero order and first order in the small variables. To obtain the generator of a z corresponding to the generator K denoted below as κ we write.36 ML µ νa ν a MA µ a.49 assuming note that g µ ν is just the familiar Kronecker δ µν L µ ν g µ ν + ɛ K µ ν.50 a + ɛ κ..5 Insertion of K µ ν as given in 0.48 yields for the l.h.s. of.49 noting the linearity of MA µ MA µ + ɛ K µ ν A ν MA µ + ɛ M A A 0 0 0 MA µ ɛ σ 0 A ɛ σ A 0.5 where we employed.5 in the last step. For the r.h.s. of.49 we obtain using.5 + ɛ κ MA µ + ɛ κ MA µ + ɛ κ MA µ + MA µ κ + Oɛ MA µ + ɛ κ σ µ + σ µ κ Aµ + Oɛ..53 Equating.5 and.53 results in the condition σ 0 A σ A 0 κ σ µ + σ µ κ Aµ..54 This reads for the four cases A µ 0 0 0 A µ 0 0 0 A µ 0 0 0 A µ 0 0 0 σ κ σ 0 + σ 0 κ κ + κ.55 σ 0 κ σ + σ κ.56 0 κ σ + σ κ.57 0 κ σ 3 + σ 3 κ..58 One can verify readily that κ σ.59

.3: Spinors 363 obeys these conditions. Similarly one can show that the generators κ κ 3 of a z corresponding to K K 3 and λ λ λ 3 corresponding to J J J 3 are given by κ σ We can hence state that the following two transformations are equivalent i λ σ..60 L w ϑ e w K + ϑ J }{{} SO3 acts on 4 vectors A µ a w i ϑ e w i ϑ σ }{{} SL C acts on spinors φ α C characterized below.6 This identifies the transformations a z w i ϑ as representations of Lorentz transformations w describing boosts and ϑ describing rotations. Exercise..4: Show that the generators.60 of a SL C correspond to the generators K and J of L µ ν..3 Spinors Definition of contravariant spinors We will now further characterize the states on which the transformation a z and its conjugate complex a z act the so-called contravariant spinors. We consider first the transformation a z which acts on a -dimensional space of states denoted by φ α φ C..6 According to our earlier discussion holds φ φ transforms under rotations z i ϑ like a spin state + φ transforms under rotations z i ϑ like a spin state. We denote the general z w + i ϑ transformation by φ α a α β φ β def a α φ + a α φ α.63 where we extended the summation convention of 4-vectors to spinors. Here a z a α a β a a a.64 describes the matrix a z.

364 Spinor Formulation Definition of a scalar product The question arises if for the states φ α there exists a scalar product which is invariant under Lorentz transformations i.e. invariant under transformations a z. Such a scalar product does indeed exist and it plays a role for spinors which is as central as the role of the scalar product A µ A µ is for 4 vectors. To arrive at a suitable scalar product we consider first only rotational transformations aiϑ. In this case spinors φ α transform like spin states and an invariant which can be constructed from products φ χ etc. is the singlet state. In the notation developed in Chapter 5 holds for the singlet state ; 0 0 0 0 m; m m m.65 m± where describes the spin state of particle and describes the spin state of particle and 0 0 ± ; stands for the Clebsch Gordon coefficient. Using 0 0 ± ; ±/ and equating the spin states of particle with the spinor φ α those of particle with the spinor χ β one can state that the quantity Σ φ χ φ χ.66 should constitute a singlet spin state i.e. should remain invariant under transformations ai ϑ. In fact as we will demonstrate below such states are invariant under general Lorentz transformations a z. Definition of covariant spinors Expression.66 is a bilinear form invariant and as such has the necessary properties of a scalar 3 product. However this scalar product is anti-symmetric i.e. exchange of φ α and χ β alters the sign of the expression. The existence of a scalar product gives rise to the definition of a dual representation of the states φ α denoted by φ α. The corresponding states are defined through φ χ φ χ φ χ + φ χ.67 It obviously holds χ α χ χ χ χ..68 We will refer to φ α χ β... as contravariant spinors and to φ α χ β... as covariant spinors. The relationship between the two can be expressed φ φ ɛ φ φ.69 0 ɛ.70 0 3 Scalar implies invariance under rotations and is conventionally generalized to invariance under other symmetry trasnformations.

.3: Spinors 365 as can be verified from.68. The inverse of.69.70 is φ φ ɛ ɛ φ φ 0 0.7..7 Exercise.3.: Show that for any non-singular complex matrix M holds ɛ M ɛ detm M T The matrices ɛ ɛ connecting contravariant and covariant spinors play the role of the metric tensors g µν g µν of the Minkowski space [cf. 0.0 0.74]. Accordingly we will express.69.70 and.7.7 in a notation analogous to that chosen for contravariant and covariant 4 vectors [cf. 0.7] The scalar product.67 will be expressed as For this scalar product holds φ α ɛ αβ φ β.73 φ α ɛ αβ φ β.74 ɛ ɛ ɛ αβ 0.75 ɛ ɛ 0 ɛ αβ ɛ ɛ 0 ɛ ɛ.76 0 φ α χ α φ χ + φ χ φ χ φ χ..77 φ α χ α χ α φ α..78 The transformation behaviour of φ α according to.63.73.75 is given by as can be readily verified. Proof that φ α χ α is Lorentz invariant φ α ɛ αβ a β γ ɛ γδ φ δ.79 We want to verify now that the scalar product.77 is Lorentz invariant. In the transformed frame holds φ α χ α a α β ɛ αγ a γ δɛ δκ φ β χ κ..80 One can write in matrix notation a α β ɛ αγ a γ δɛ δκ [ ɛ a ɛ T a ] κβ..8

366 Spinor Formulation Using. and.4 one can write ɛ a ɛ ɛ e z σ ɛ e z ɛ σɛ e z σ.8 and with fa T fa T for polynomial fa ɛ a ɛ T e z σ T e z σ a.83 Here we have employed the hermitian property of σ i.e. σ σ T σ. Insertion of.83 into.8 yields a α β ɛ αγ a γ δɛ δκ [ ɛ a ɛ T a ] κβ [ a a ] κβ δ κβ.84 and hence from.80 φ α χ α φ β χ β..85 The complex conjugate spinors We consider now the conjugate complex spinors φ α φ φ..86 A concise notation of the conjugate complex spinors is provided by φ α φ α φ φ.87 which we will employ from now on. Obviously it holds φ k φ k k. The transformation behaviour of φ α is φ α a α β φ β.88 which one verifies taking the conjugate complex of.63. As discussed above a z provides a representation of the Lorentz group which is distinct from that provided by a z. Hence the conjugate complex spinors φ α need to be considered separately from the spinors φ α. We denote a α β a α β.89 such that.88 reads φ α a α βφ β.90 extending the summation convention to dotted indices. We also define covariant versions of φ α φ α φ α..9

.4: Spinor Tensors 367 The relationship between contravariant and covariant conjugate complex spinors can be expressed in analogy to.73.76 φ α ɛ α β φ β.9 φ α ɛ α β φ β.93 ɛ α β ɛ αβ.94 α β ɛ ɛ αβ.95 where ɛ αβ and ɛ αβ are the real matrices defined in.75.75. For the spinors φ α and χ α thus defined holds that the scalar product is Lorentz invariant a property which is rather evident. The transformation behaviour of φ α is φ α χ α φ χ + φ χ.96 φ α ɛ α βa β γ ɛ γ δφ δ..97 The transformation in matrix notation is governed by the operator ɛ a z ɛ which arises in the Lorentz transformation.8 of the bispinor wave function Ψ ɛ a z ɛ accounting for the transformation behaviour of the third and fourth spinor component of Ψ. A comparision of.8 and.97 implies then that φ α transforms like Ψ 3 Ψ 4 i.e. one can state φ transforms under rotations z i ϑ like a spin state + φ transforms under rotations z i ϑ like a spin state. The transformation behaviour of Ψ note that we do not include presently the space-time dependence of the wave function Ψ S z Ψ a z Ψ Ψ ɛa zɛ.98 Ψ3 Ψ 4 obviously implies that the solution of the Dirac equation in the chiral representation can be written in spinor form Ψ x µ φ x µ Ψ x µ Ψ 3 x µ φ x µ φ α χ xµ x µ χ βx µ..99 Ψ 4 x µ χ xµ.4 Spinor Tensors We generalize now our definition of spinors φ α to tensors. A tensor t α α α k β β β l.00

368 Spinor Formulation is a quantity which under Lorentz transformations behaves as t α α α k β β β l k m a α m γm l a β n δn t γ γ k δ δ l..0 n An example is the tensor t α β which will play an important role in the spinor presentation of the Dirac equation. This tensor transforms according to α β t a α γa β δ t γ δ.0 This reads in matrix notation using conventional matrix indices j k l m t jk a t a a jl a km t lm..03 jk lm Similarly the transformation bevaviour of a tensor t αβ reads in spinor and matrix notation t αβ a α γ a β δ t γδ t jk a t a T jk lm a jl a km t lm.04 Indices on tensors can also be lowered employing the formula t α β ɛ αγ t γ β.05 and generalizations thereof. An example of a tensor is ɛ αβ and ɛ αβ. This tensor is actually invariant under Lorentz transformations i.e. it holds ɛ αβ ɛ αβ ɛ αβ ɛ αβ.06 Exercise.4.: Prove equation.06. The 4 vector A µ in spinor form We want to provide now the spinor form of the 4-vector A µ i.e. we want to express A µ through a spinor tensor. This task implies that we seek a tensor the elements of which are linear functions of A µ. An obvious candidate is [cf..5] MA µ σ µ A µ. We had demonstrated that MA µ transforms according to M ML µ νa ν a MA µ a.07 which reads in spinor notation [cf..0.03 A α β a α γa β δa γ δ..08 Obviously this transformation behaviour is in harmony with the tensor notation adopted i.e. with contravariant indices α β. According to.5 the tensor is explicitly A α β A A A 0 + A 3 A ia A + ia A 0 A 3..09 A A

.4: Spinor Tensors 369 One can express A α β also through A µ A α β A0 A 3 A + ia A ia A 0 + A 3 The 4-vectors A µ A µ can also be associated with tensors..0 A α β ɛ αγ ɛ β δa γ δ.. This tensor reads in matrix notation A A A A 0 0 A A A A A A A A 0 0.. Hence employing.09.0 one obtains A α β A 0 A 3 A ia A + ia A 0 + A 3.3 A α β A0 + A 3 A + ia A ia A 0 A 3..4 We finally note that the 4 vector scalar product A µ B µ reads in spinor notation A µ B µ Aα βb α β..5 Exercise.4.: Prove that.5 is correct. µ in spinor notation The relationship between 4 vectors A µ A µ and tensors t α β can be applied to the partial differential operator µ. Using.0 one can state α β 0 3 + i i 0 + 3..6 Similarly.4 yields α β 0 + 3 + i i 0 3..7

370 Spinor Formulation σ µ in Tensor Notation We want to develop now the tensor notation for σ µ.6 and its contravariant analogue σ µ σ µ σ µ 0 0 0 0 0 0 0 0 0 i i 0 0 i i 0 0 0 0 0.8 For this purpose we consider first the transformation behaviour of σ µ and σ µ. We will obtain the transformation behaviour of σ µ σ µ building on the known transformation behaviour of γ µ. This is possible since γ µ can be expressed through σ µ σ µ. Comparision of 0.9 and.8 yields γ µ 0 σ µ..9 σ µ 0 Using.5 one can write and hence γ µ σ µ ɛ σ µ ɛ.0 0 σ µ ɛ σ µ ɛ 0.. One expects then that the transformation properties of σ µ should follow from the transformation properties established already for γ µ [c.f. 0.43]. Note that 0.43 holds independently of the representation chosen i.e. holds also in the chiral representation. To obtain the transformation properties of σ µ we employ then 0.43 in the chiral representation expressing SL η ξ by.8 and γ µ by.. Equation 0.43 reads then a 0 0 ɛa ɛ 0 σ µ ɛσ µ ɛ 0 a 0 0 ɛa ɛ L ν µ 0 σ ν ɛσ ν ɛ 0.. The l.h.s. of this equation is 0 aσ µ ɛ a ɛ ɛa σ µ ɛ a 0 L ν µ.3 and hence one can conclude Equation.5 is equivalent to aσ µ ɛ a ɛ L ν µ σ ν.4 ɛa σ µ ɛ a L ν µ ɛσ ν ɛ..5 a σ µ ɛ a ɛ L ν µ σ ν.6 which is the complex conjugate of.4 i.e..5 is equivalent to.4. Hence.4 constitutes the essential transformation property of σ µ and will be considered further.

.4: Spinor Tensors 37 One can rewrite.4 using.6. ɛ a ɛ ɛ e z σ ɛ e z ɛ σ ɛ e z σ..7 Exploiting the hermitian property of σ e.g. σ T σ yields using. ɛ a ɛ e z σ T One can express therefore equation.4 [ e z σ ] T [a ] T..8 a σ µ [a ] T L ν µ σ ν..9 We want to demonstrate now that the expression aσ µ [a ] T is to be interpreted as the transform of σ µ under Lorentz transformations. In fact under rotations the Pauli matrices transform like j 3 σ j aiϑ σ j ai ϑ aiϑ σj a iϑ T..30 We argue in analogy to the logic applied in going from.07 to.08 that the same transformation behaviour applies then for general Lorentz transformations i.e. transformations..4 with w 0. One can hence state that σ µ in a new reference frame is σ µ a σ µ a.3 where a is given by..4. This transformation behaviour according to.0.03 identifies σ µ as a tensor of type t α β. It holds according to.8 σ µ σ µ σ µ σ µ 0 0 0 i 0 0 0 i 0 0 }{{}}{{}}{{}}{{} µ 0 µ µ µ 3.3 and 0 0 }{{} µ 0 σµ σ µ σ µ σ µ 0 0 i 0 0 i 0 0 }{{}}{{}}{{} µ µ µ 3..33 Combining.9.3 one can express the transformation behaviour of σ µ in the succinct form L ν µ σ µ σ ν..34

37 Spinor Formulation Inverting contravariant and covariant indices one can also state L ν µ σ µ σ ν..35 This is the property surmised already above [cf..9]. We can summarize that σ µ and σ µ transform like a 4 vector however the transformation is inverse to that of ordinary 4 vectors. Each of the 4 4 6 matrix elements in.3 and.33 is characterized through a 4-vector index µ µ 0 3 as well as through two spinor indices α β. We want to express now σ µ and σ µ also with respect to the 4 vector index µ in spinor form employing.4. This yields σ α β σ0 + σ 3 σ + iσ σ iσ σ 0 σ 3 σ α β γ δ 0 0 0 0 0 0 0 0 0 0 0 0 where on the rhs. the submatrices correspond to t α β spinors. We can in fact state σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ Equating this with the r.h.s. of.36 results in the succinct expression.36..37 σ α β γ δ δαγ δ β δ..38 Note that all elements of σ α β are real and that there are only four non-vanishing elements. In.38 the inner covariant spinor indices i.e. α β account for the 4 vector index µ whereas the outer contravariant spinor indices. i.e. γ δ account for the elements of the individual Pauli matrices. We will now consider the representation of σ µ σ µ in which the contravariant indices are moved inside i.e. account for the 4-vector µ and the covariant indices are moved outside. The desired change of representationσ µ α β σ µ α β corresponds to a transformation of the basis of spin states f g g f ɛ f g and hence corresponds to the transformation σµ σ µ 0 σ µ σ µ σ µ σ µ 0 σ µ σ µ 0 0.39.40 where we employed the expressions. for ɛ and ɛ. Using.0 to express σ α β in terms of σ µ yields together with 5.4 0 0 0 0 σ α β σ 0 σ 3 σ + iσ 0 0 σ iσ σ 0 + σ 3.4 0 0 0 0 0

.4: Spinor Tensors 373 and employing then transformation.40 to transform each of the four submatrices which in.37 are in a basis α β to a basis α β results in σ α β γ δ This can be expressed σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ 0 0 0 0 0 0 0 0 0 0..4 0 0 σ α β γ δ δ αγδ β δ..43 Combined with.38 one can conclude that the following property holds γ δ σ α β σ α β γ δ δ αγδ β δ..44 The Dirac Matrices γ µ in spinor notation We want to express now the Dirac matrices γ µ in spinor form. For this purpose we start from the expression. of γ µ. This expression implies that the element of γ µ given by ɛ σ µ ɛ is in the basis α β whereas the element of γ µ given by σ µ is in the basis α β. Accordingly we write γ µ 0 σ µ α β σ µ α β 0..45 Let A µ be a covariant 4 vector. One can write then the scalar product using.5 γ µ A µ 0 σ µ α βa µ σ µ α β A µ 0 0 σ βa γ δ α γ δ σγ α β A γ δ 0 Exploiting the property.44 results in the simple relationship γ µ A µ 0 A α β A α β 0..46..47

374 Spinor Formulation.5 Lorentz Invariant Field Equations in Spinor Form Dirac Equation.47 allows us to rewrite the Dirac equation in the chiral representation 0.6 i γ µ µ Ψx µ 0 i α β Ψx µ m i α β 0 Ψx µ..48 Employing Ψx µ in the form.99 yields the Dirac equation in spinor form The simplicity of this equation is striking. i α β χ β m φ α.49 i α β φ α m χ β..50