Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory Rainer Kühne

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Rainer Kühne. Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory. 2018. hal-01936246

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Derivation of the Local Lorentz Gauge Transformation of a Dirac Spinor Field in Quantum Einstein-Cartan Theory

Rainer W. Kühne Tuckermannstr. 35, 38118 Braunschweig, Germany e-mail: [email protected] July 2, 2018

Abstract Hehl et al. [6] (Einstein-Cartan theory), Mes- siah [7] (non-relativistic quantum mechanics), I examine the groups which underly classical Bjorken and Drell [8] (relativistic quantum me- mechanics, non-relativistic quantum mechanics, chanics), Bjorken and Drell [9] (quantum elec- special relativity, relativistic quantum mechan- trodynamics), Taylor [10] (quantum flavourdy- ics, quantum electrodynamics, quantum flavour- namics), Politzer [11] and Marciano and Pagels dynamics, quantum chromodynamics, and gen- [12] (quantum chromodynamics), and Kühne eral relativity. This examination includes the [13] (quantum electromagnetodynamics). rotations SO(2) and SO(3), the Pauli algebra, the Lorentz transformations, the Dirac algebra, and the U(1), SU(2), and SU(3) gauge trans- 2 Classical Mechanics formations. I argue that general relativity must be generalized to Einstein-Cartan theory, so that The space of classical mechanics is described Dirac spinors can be described within the frame- by the three-dimensional Euclidian space. The work of gravitation theory. scalar product of the three-vectors ai and bi is given by a · b = δ a b (2) Keywords ij i j Einstein-Cartan theory, quantum gravity, where energy-momentum tensor, Lorentz transfor- δij = diag (1, 1, 1) (3) mations, gauge invariance, quantum field denotes the Kronecker symbol. The vector prod- theories uct is given by

1 Introduction (a × b)k = εijkaibj (4) where εijk denotes the totally anti-symmetric I will present my argumentation in a didactic Levi-Civita symbol. The square of the infinites- way. So I will start at a very elementary level. imal line element is given by If not denoted otherwise then Latin indices run 2 from 1 to 3, Greek indices run from 0 to 3. I will ds = δijdxidxj (5) use the Einstein summation convention, where it is summed over all indices which appear twice. where dxi denotes the infinitesimal coordinate Moreover I will use the natural units difference of the two space-points xi and yi

¯h = c = ε0 = 1 (1) dxi = lim (yi − xi) (6) yi→xi where ¯h = h/2π denotes the reduced Planck con- A rotation R around the rotation angle ϕ in the stant, c the speed of light, and ε0 the electric ﬁeld constant. Inner indices of matrices will be two-dimensional subspace is described by the or- dropped. thogonal rotation group SO(2) Readers who would like to learn more about R = exp(−iϕD) the theories which will be mentioned in this cos ϕ − sin ϕ paper are recommended to the comprehensive = ∈ SO(2) (7) works of Goldstein [1] (classical mechanics), sin ϕ cos ϕ Jackson [2] (classical electrodynamics), Kilmis- where, because of the Euler equation ter [3] (special relativity), Weinberg [4] and Mis- ner, Thorne and Wheeler [5] (general relativity), e−iϕ = cos ϕ − i sin ϕ (8)

1 the generator of the rotation is SU(2) is covered by the group SO(3). Note also that the commutators eq. (11) and eq. (19) dif- 0 −i D = (9) fer by a factor of two. The anti-commutator is i 0

{σi, σj} = 2δij (20) A rotation R around the rotation angle ϕi in the three-dimensional space is described by the The multiplication of two three-vectors ai and rotation group SO(3) bi is given by

R = exp(−iϕiDi) ∈ SO(3) (10) (σ · a)(σ · b) = (σi · ai)(σj · bj) where the generators Di of the rotation satisfy = (δij + iεijkσk)aibj the commutator relation = a · b + iσ · (a × b) (21)

[Di,Dj] = iεijkDk (11) According to non-relativistic quantum mechanics both spin and isospin are invariant under A representation of the generators of the SO(3) global transformations of the group SU(2). If group is Ψ denotes a two-component Pauli-spinor, ϕi the  0 0 0  three-component rotation angle vector, σi the Pauli matrices, and x a space-time point, then D1 =  0 0 −i  (12) 0 i 0 Ψ0(x) = exp(−iϕiσi/2)Ψ(x) (22)  0 0 i 

D2 =  0 0 0  (13) If Ψ denotes a two-component iso-spinor, ϕi the −i 0 0 three-component phase vector, and τi the Pauli  0 −i 0  matrices, then D3 =  i 0 0  (14) 0 0 0 Ψ0(x) = exp(−iϕiτi/2)Ψ(x) (23)

Note that the number of generators of the or- thogonal groups is given by 4 Special Relativity

dim SO(n) = n(n − 1)/2 (15) The special theory of relativity is invariant under the semi-simple Poincare group. The pa- and that the generators of SO(n) are Hermitean. rameters of its translational part are time and three-position. The parameters of its rotational part are the rotation angle three-vector and the 3 Non-Relativistic Quantum three-component Lorentz boost. Mechanics The scalar product of the two four-vectors aµ and bµ is given by Classical angular momentum is continuous. In µ ν quantum physics orbital angular momentum is a · b = gµν a b (24) quantized in units of ¯h and intrinsic spin is quantized in units of ¯h/2. Intrinsic spin is described where gµν denotes the metric tensor. It is by the unitary group SU(2). Its generators are gµν = g (25) the three Pauli matrices σi. An often used rep- µν µ µ µ resentation of the Pauli matrices is g ν = gν = δν = diag (1, 1, 1, 1) (26) 0 1 In Minkowski coordinates the metric tensor is σ1 = (16) 1 0 represented by 0 −i σ2 = (17) i 0 gµν = diag (1, −1, −1, −1) (27) 1 0 σ3 = (18) The square of the infinitesimal line element is 0 −1 given by 2 µ ν The commutator of the group SU(2) is given by ds = gµν dx dx (28) where dxµ denotes the infinitesimal coordinate [σ , σ ] = 2iε σ (19) i j ijk k difference of the two space-time points xµ and µ The groups SU(2) and SO(3) are local isomor- y dxµ = lim (yµ − xµ) (29) phic for angles 0 ≤ ϕ < 2π where the group yµ→xµ

2 A rotation around the z-axis by the rotation an- 5 Relativistic Quantum Me- gle ϕ is given by chanics  1 0 0 0  µ  0 cos ϕ − sin ϕ 0  The group which underlies the kinematics of a ν =   (30)  0 sin ϕ cos ϕ 0  relativistic quantum mechanics and relativistic 0 0 0 1 quantum field theory is the Poincare group. Isospin is invariant under global transformations A Lorentz boost along the x-axis by the speed v of the group SU(2). The generators are the is given by three Pauli matrices τ . The three Pauli matri-   i γ −βγ 0 0 ces σi of spin are generalized by the four Dirac µ  −βγ γ 0 0  matrices γµ. An often used representation of the a ν =   (31)  0 0 1 0  Dirac matrices is 0 0 0 1 1 0 γ = (43) where 0 0 −1 β = v/c (32) 0 σ γ = i (44) i −σ 0 γ = 1/p1 − v2/c2 (33) i µ It is In general, a Lorentz transformation a ν of a four-position xµ is given by 0 γ = γ0 (45) µ µ ν i x0 = a ν x (34) γ = −γi (46) The Lorentz transformation of a four-derivative The commutator is µ ∂µ = ∂/∂x is given by ν [γµ, γν ] = −2iσµν (47) ∂0µ = aµ ∂ν (35) which is the definition of the generalized Dirac Finally, a Lorentz transformation around the pa- matrices σµν . The anti-commutator gives rameter ωαβ is given by µ {γµ, γν } = 2gµν (48) µ 1 αβ a ν = exp ωαβI ∈ O(1, 3) (36) 2 ν By using the representation eqs. (43) and (44) where a representation of the six generators of of the Dirac matrices and the representation the Lorentz group SO(1, 3) is eqs. (16), (17) and (18) of the Pauli matrices, the anti-commutator gives the representation eq.   0 −1 0 0 (27) of the metric tensor in Minkowski coordi- 1 0 0 0 I10 =   (37) nates. Moreover it is  0 0 0 0    5 0 0 0 0 γ = γ5 = −iγ0γ1γ2γ3 (49)   0 0 −1 0 The multiplication of two four-vectors aµ and bµ 0 0 0 0 I20 =   (38) is given by  1 0 0 0    µ ν 0 0 0 0 (γ · a)(γ · b) = (γµa )(γν b ) µ ν  0 0 0 −1  = (gµν − iσµν )a b (50) 30  0 0 0 0  The Lorentz transformation of a four-component I =   (39)  0 0 0 0  spin 1/2 Dirac spinor field Ψ(x) is given by 1 0 0 0 i 1 µν  0 0 0 0  Ψ0(x0) = exp − σ ω Iαβ Ψ(x) 4 µν 2 αβ 0 0 0 −1 I13 =   (40) (51)  0 0 0 0    which is a generalization of eq. (36) which con- 0 1 0 0 siders both the Lorentz transformation of the   0 0 0 0 elementary particle and its intrinsic spin. 23  0 0 0 0  I =   (41)  0 0 0 −1  0 0 1 0 6 Relativistic Quantum  0 0 0 0  Field Theory 12  0 0 −1 0  I =   (42)  0 1 0 0  Quantum electrodynamics is invariant under lo- 0 0 0 0 cal transformations of the gauge group U(1).

3 If Ψ describes a four-component Dirac spinor, A representation of the Gell-Mann matrices is x a space-time point, e the elementary electric  0 1 0  charge, ϕ the gauge phase, Dµ the covariant λ1 = 1 0 0 (61) derivative, ∂µ the partial four-derivative, and   Aµ the electromagnetic four-potential, then the 0 0 0 gauge transformation is  0 −i 0  λ2 =  i 0 0  (62) 0 0 0  1 0 0  Ψ0(x) = exp(−ieϕ(x))Ψ(x) (52) λ3 =  0 −1 0  (63) iDµ = i∂µ − eAµ(x) (53) 0 0 0 A0µ(x) = Aµ(x) − ∂µϕ(x) (54)  0 0 1  λ4 =  0 0 0  (64) 1 0 0   Note that U(1) and SO(2) are isomorphic. 0 0 −i λ5 =  0 0 0  (65) i 0 0 Quantum ﬂavourdynamics is invariant un-   der local transformations of the gauge group 0 0 0 SU(2) × U(1). If Ψ denotes an eight-component λ6 =  0 0 1  (66) iso-spinor Dirac spinor, x a space-time point, 0 1 0 g the weak coupling constant, ϕi the three-  0 0 0  component gauge phase isovector, τ the Pauli i λ7 = 0 0 −i (67) i   matrices, and Wµ the weak isovector four- 0 i 0 potentials, then the gauge transformations of the  1 0 0  SU(2) part are 1 λ8 = √  0 1 0  (68) 3 0 0 −2 The commutator of the Gell-Mann matrices is Ψ0(x) = exp(−igϕi(x)τi/2)Ψ(x) (55) g [λ , λ ] = 2if λ (69) iD = i∂ − W i (x)τ (56) i j ijk k µ µ 2 µ i µ µ µ and the anti-commutator is W 0i (x) = Wi (x) − ∂ ϕi(x) µ 4 −gεijkϕj(x)Wk (x) (57) {λ , λ } = δ + 2d λ (70) i j 3 ij ijk k Note that the number of the generators of the unitary groups is given by

Quantum chromodynamics is invariant un- dim SU(n) = n2 − 1 (71) der local transformations of the gauge group SU(3). If Ψ denotes a twelve-component colour- and that the generators of the SU(n) groups are vector Dirac spinor, x a space-time point, g Hermitean. the strong coupling constant, ϕ the eight- According to relativistic quantum mechan- i ics and relativistic quantum ﬁeld theory, the component gauge phase vector, λi the eight Gell- µν i energy-momentum tensor Σ of a Dirac spinor Mann matrices, and Gµ the eight gluon four- potentials, then the gauge transformations are Ψ is asymmetric 1 Σµν (x) = − DµΨ(¯ x) γν Ψ(x) 2 1 + Ψ(¯ x)γν DµΨ(x) (72) Ψ0(x) = exp(−igϕi(x)λi/2)Ψ(x) (58) 2 g iD = i∂ − Gi (x)λ (59) µ µ 2 µ i where the covariant derivative Dµ is given by µ µ µ the equations (53), (56) and (59). G0i (x) = Gi (x) − ∂ ϕi(x) µ −gfijkϕj(x)Gk (x) (60) 7 General Relativity

The general theory of relativity is invariant un- where the indices i, j, k run from 1 to 8. der arbitrary curvilinear transformations. It

4 is invariant under local transformations of the Since the affine connection (Christoffel sym- Lorentz group SO(1, 3). The Lorentz boosts de- bol) is symmetric it follows that the energy- pend on the four-position xµ. Note that the momentum tensor of general relativity is sym- Poincare group which underlies special relativity metric. This is in contrast to the asymmet- has ten generators, whereas the Lorentz group ric energy-momentum tensor of a Dirac spinor which underlies general relativity has only six of relativistic quantum mechanics. This means generators. Therefore general relativity is not a that a Dirac spinor cannot be described by the generalization of special relativity. geometry that underlies general relativity. The scalar product of the four-vectors aµ and bµ is given by 8 Einstein-Cartan Theory µ ν a · b = gµν (x)a b (73) Einstein-Cartan theory is a generalization of where the metric tensor gµν depends on the general relativity, because within the framework space-time point x. The square of the infinites- of this theory the anti-symmetric part of the imal line element is affine connection which is named Cartan’s torsion tensor 2 µ ν ds = gµν (x)dx dx (74) 1 T α (x) = Γα (x) − Γα (x) (83) The local Lorentz transformation is µν 2 µν νµ µ µ 1 αβ is nonzero. In contrast to the Christoffel symbol a ν (x) = exp ωαβ(x)I (75) 2 ν the torsion tensor transforms as a tensor under arbitrary curvilinear transformations. Cartan’s which is a generalization of eq. (36). torsion tensor is related to the spin tensor by When a four-vector Cα is parallely displaced from the four-position xµ to the four-position T αµν (x) = κτ αµν (x) (84) xµ + dxµ, then it changes according to the pre- scription where κ = −8πG denotes the Einstein field constant known from general relativity. α α ν µ dC = −Γµν (x)C dx (76) The Dirac spinor can be described by the geometry that underlies Einstein-Cartan theory, This is the definition of the four-position- α because this theory describes an asymmetric dependent affine connection Γµν . According to affine connection and therefore an asymmetric general relativity it has only a symmetric part energy-momentum tensor. 1 Einstein-Cartan theory is invariant under lo- {}α (x) = Γα (x) + Γα (x) (77) µν 2 µν νµ cal transformations of the Poincare group. So Einstein-Cartan theory is a generalization of which is named Christoffel symbol. The Rie- special relativity. mann curvature tensor is given by In quantum Einstein-Cartan theory the local Lorentz gauge transformation of a Dirac spinor Rλ (x) = ∂ Γλ (x) − ∂ Γλ (x) µνκ κ µν ν µκ field Ψ is given by α λ α λ +Γµν (x)Γκα(x) − Γµκ(x)Γνα(x) i 1 µν (78) Ψ0(x0) = exp − σ ω (x)Iαβ Ψ(x) 4 µν 2 αβ By contraction one gets the Ricci tensor (85) This equation is a combination of eqs. (51) and λ Rµν (x) = R µλν (x) (79) (75). and the Ricci scalar

µ 9 Conclusion R(x) = Rµ(x) (80) I argued that the general theory of relativity The Einstein field equations are must be generalized by the Einstein-Cartan the- 1 ory, because the asymmetric energy-momentum R (x) − g (x)R(x) = κΣ (x) (81) µν 2 µν µν tensor of relativistic quantum mechanics re- quires the existence of an asymmetric affine con- where nection and therefore the existence of a nonzero κ = −8πG (82) torsion tensor. Moreover I pointed out that spin denotes the Einstein field constant and Σµν the is the source of a gauge field which is associ- energy-momentum tensor. ated with Cartan’s torsion. This is analogous to

5 the situation of isospin which is a pure quantum number in quantum mechanics, but the source of the weak gauge ﬁeld in quantum ﬂavourdy- namics.

References

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