Fermionic one-particle states in curved spacetimes

Journal of High Energy Physics, Jul 2018

Abstract We show that a notion of one-particle state and the corresponding vacuum state exists in general curved backgrounds for spin \( \frac{1}{2} \) fields. A curved spacetime can be equipped with a coordinate system in which the metric component g−− = 0. We separate the component of the left-handed massless Dirac field which is annihilated by the null vector ∂− and compute the corresponding Feynman propagator. We find that the propagating modes are localized on two dimensional subspaces and the Feynman propagator is similar to the Feynman propagator of chiral fermions in two dimensional Minkowski spacetime. Therefore, it can be interpreted in terms of one-particle states and the corresponding vacuum state similarly to the second quantization in Minkowski spacetime.

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Fermionic one-particle states in curved spacetimes

HJE Fermionic one-particle states in curved spacetimes Farhang Loran 0 0 Department of Physics, Isfahan University of Technology We show that a notion of one-particle state and the corresponding vacuum state exists in general curved backgrounds for spin 12 fields. A curved spacetime can be equipped with a coordinate system in which the metric component g−− = 0. We separate the component of the left-handed massless Dirac field which is annihilated by the null vector ∂− and compute the corresponding Feynman propagator. We find that the propagating modes are localized on two dimensional subspaces and the Feynman propagator is similar to the Feynman propagator of chiral fermions in two dimensional Minkowski spacetime. Therefore, it can be interpreted in terms of one-particle states and the corresponding vacuum state similarly to the second quantization in Minkowski spacetime. Nonperturbative Effects; Field Theories in Lower Dimensions 1 Introduction 2 General properties of Feynman propagator 2.1 Massless Dirac field 3.1 Upside down approach 4.1 4.2 4.3 Feynman propagator Curvature Left-handed frames 3 Two dimensional Minkowski spacetime 4 Two dimensional curved spacetime 5 Four dimensional curved spacetime 5.1 Conformally flat spacetimes 6 One particle states 6.1 The Kerr solution 7 Conclusion 1 Introduction Z := Z DψDψeiS . – 1 – The study of massless Dirac fermions on non-stationary curved backgrounds has diverse theoretical and observational motivations [1–6]. Although the concepts of the vacuum state and the one-particle states are at the core of the second quantization in Minkowski spacetime, in general, they are not well-defined in quantum field theory in time-dependent curved spacetimes [7]. In principle, the vacuum state can be inferred from a plausible two-point function of the quantum fields [8]. Such a two-point function can be computed by path integrals. For Dirac spinors it is given by SF (x, x′) = Z −1 Z DψDψeiS ψ(x)ψ(x′), where ψ and ψ are Grassmann fields, S denotes the action and Eq. (1.1) implies that SF (x, x′) is a Green’s function for Dirac operator, i.e., it solves Dirac equation with the Dirac delta-function source, satisfying certain boundary conditions. where v(p)eip·x and u(p)e−ip·x are solutions of Dirac field equation, with positive energy and negative energy respectively, and A and B denote other quantum numbers collectively. For example, for fermions coupled to Maxwell field, A and B denote the definition ψ := ψ†γ0. Eqs. (1.6) and (1.7) give, respectively, the electric charges q and −q respectively. The expressions for ψ(x) and ψ(x) follows represent the propagation of positive-energy particles from x′ to x and from x to x′ respecHJEP07(218) tively. That is, we suppose that3 s s s s In four dimensional Minkowski spacetime, Dirac operator in an inertial (nonaccelerating) reference frame is (i∂/ − m),1 and SF (x, x′) equals the Feynman propagator2 SF (x, x′) = SA(x, x′)θ(x0 − x′0) − SB(x, x′)θ(x′0 − x0), in which, the amplitudes SA(x, x′) := Dψ(x) ψ(x′)E , SB(x, x′) := Dψ(x′) ψ(x)E , (1.3) (1.4) (1.5) (1.6) (1.7) (1.8) (1.9) (1.10) (1.11) (1.12) (1.13) (1.14) (1.15) Lorentz transformation and the field equations (1.8) and (1.9) give ψ(x) = X Z ψ(x) = X Z d3p d3p (2π)3p2Ep (2π)3p2Ep v¯s(p)e−ip·x hp~, s; B| , u¯s(p)eip·x |p~, s; Ai . s s X vs(p)v¯s(p) = p/ − m, X us(p)u¯s(p) = p/ + m, Dp~, s; A ~q, r; AE = (2π)3δ3(p~ − ~q)δrs, Dp~, s; B ~q, r; BE = (2π)3δ3(p~ − ~q)δrs. where we have also fixed the normalization constants. Translational and rotational invariance imply that4 [9] 1m denotes the mass of Dirac particle, ∂/ := γa∂a and γa are Dirac matrices. 2θ denotes the Heaviside step function. θ(x) equals 1 and 0, for x > 0 and x < 0 respectively. 3Ep := pp~2 + m2. p · x := ηabpapb where p0 := Ep, and the Minkowski metric ηab = diag( 1, −1, −1, −1 ). 4δD denotes the Dirac delta-function in D-dimensions. The Kronecker delta δrs equals 1 and 0 for r = s and r 6= s respectively. – 2 – we obtain The spinors anti-commute at spacelike separation, i.e., SA(x, x′) = SB(x, x′) = Z Z d3p p/ + m (2π)3 2Ep d3p p/ − m eip·(x−x′). SA(x, x′) = −SB(x, x′), such that g−− = 0. In a frame of reference given by7 tetrad identifying the frame of reference. ∂(a) := eμ(a)∂μ. 5A counterexample is the non-inertial frame corresponding to Rindler observers in Minkowski spacetime because the Rindler wedge of Minkowski spacetime is globally hyperbolic [11]. 6There are also nonlocal effects such as the appearance of event horizons in the accelerating frames, that add nontrivial features to such descriptions. For example a uniformly accelerated observer in Minkowski spacetime has access only to a subset of physical states which correspond to the Rindler wedge of the Minkowski spacetime [12]. 7 ∇μ denotes the Levi-Civita connection and g is the determinant of the spacetime metric. eμ(a) is the This in turn indicates that the particles represented by Dirac field obey Fermi statistics [9]. formation, and ladder operators arp and brp such that That is, if we postulate a vacuum state |0i invariant under translation and Lorentz transarp |0i = 0, |p~, r; Ai = arp† |0i , brp |0i = 0, |p~, r; Bi = brp† |0i , arp†aqs† |0i = −aqs†arp† |0i , brp†bqs† |0i = −bqs†brp† |0i . In general, the concept of positive energy is lost in non-stationary curved spacetime and also in non-inertial reference frames in flat spacetime5 [10]. As we have seen, this concept is essential for defining Dirac particles in inertial reference frames in Minkowski spacetime. Related to this fact, we do not have, in general, a well-defined concept of timeordering which has been used in eq. (1.3), although the Feynman propagator SF (x, x′) itself is well-defined through eq. (1.1). In principle, physical observables in accelerating frames in Minkowski spacetime can be described in terms of their counterparts in inertial frames by applying the corresponding local Lorentz transformations.6 In this paper we study massless spinors on curved spacetimes. In general, a four dimensional curved spacetime can be equipped with a coordinate system (x±, x⊥), where ∂(−) = (−g)− 21 ∂−, eν ( 1 )∇−eν( 2 ) = 0, – 3 – (1.16) (1.17) (1.18) (1.19) (1.20) (1.21) (1.22) (1.23) (1.24) (1.25) (1.26) we separate the spin-up component of the left-handed massless Dirac field along the third direction and compute the corresponding Feynman propagator S↑(L)(x, x′). We find that F S↑(L)(x, x′) = SF+(x, x′)δ2(x⊥ − x′ ), F ⊥ where SF+(x, x′) denotes the Feynman propagator of left-moving massless fermions in an inertial frame in a two dimensional Minkowski spacetime equipped with coordinates (x0, x3),8 1 2 Therefore S↑(L)(x, x′) can be interpreted in terms of one-particle states and the corresponding vacuum state, similarly to the second quantization in two dimensional Minkowski spacetime. Eq. (1.27) shows that the corresponding modes are localized on the x indeed. The sign function in eq. (1.28) can be equivalently given in terms of x− because HJEP07(218) Since x− is singled out by its definition via g−− = 0, this substitution leads to a covariant description of the Feynman propagator. Our paper is organized as follows. In section 2, we review the general properties of the Feynman propagator given by the path integral (1.1). Section 3 is devoted to review the second quantization of Dirac field in inertial frames in two dimensional flat spacetime. Our goal in section 3.1 is to explain how the concepts of vacuum state and one-particle states can be inferred from the short distance singularity of the Feynman propagator. In section 4, we compute the Feynman propagator in a two dimensional curved background. Obviously, the result depends on the frame of reference. In section 4.3, we introduce the left-handed frame in which the curvature effect is removed from the left-mover sector and the Feynman propagator is given by eq. (1.28). Following our discussion in section 3.1, we conclude that even in a non-stationary spacetime, left-mover vacuum state and left-mover one-particle states are well-defined in the left-handed frame. Of course, by removing the curvature effect from the left-handed sector, the whole effect of the spacetime curvature transmits to the right-handed sector, thus we do not expect a universally consistent definition of right-mover one-particle states. We study the curvature effect in four dimensions in section 5. In section 6 we introduce a diffeomorphism invariant action with local Lorentz symmetry for the separated spin-up component of free left-handed fermions and derive eq. (1.27). Eq. (1.29) is explained in section 6.1 where we look in on the Kerr geometry. Our results are summarized in section 7. 2 General properties of Feynman propagator Our starting point is eq. (1.1) in which (1.27) (1.28) ⊥ plane (1.29) S := i Z dDyp−g(y) ψ(y)D(y)ψ(y), (2.1) 8sgn(x) := θ(x) − θ(−x) is the sign function. – 4 – where D(y) denotes the Dirac operator. This equation implies that Noting that the action S is real-valued, eq. (2.1) also gives9 D(x)ψ(x) = − √ i −g δψ(x) δS . S := −i Z dDyp−g(y) ψ(y)γ0D←−(y−)−† γ0ψ(y), Eq. (2.3) gives Eq. (2.2) and eq. (1.1) together give ψ(x)γ0D←−(x−)−†γ0 = − √ i −g δψ(x) δS . where we have used the complex conjugation rule for Grassmann numbers = (−g(x))− 12 δD(x − x′). Eq. (2.7) is obtained by integration by part. Similarly, eq. (2.5) and eq. (1.1) give This result together with eq. (2.7) implies that So, D(x′)γ0SF (x, x′)†γ0 = −(−g(x′))− 12 δD(x − x′). D(x′) hSF (x′, x) + γ0SF (x, x′)†γ0i = 0. SF (x′, x) + γ0SF (x, x′)†γ0 ⊜ 0, where ⊜ indicates that a function U(x′) solving the homogenous field equation D(x′)U(x′) = 0 has been dropped from the right hand side of eq. (2.10). To recognize this result in the four dimensional Minkowski spacetime we note that eqs. (1.16) and (1.17) give γ0SA(x, x′)†γ0 = SA(x′, x), γ0SB(x, x′)†γ0 = SB(x′, x). Consequently, eq. (1.3) gives SF (x′, x) + γ0SF (x, x′)†γ0 = SA(x′, x) − SB(x′, x). 9γa† = γ0γaγ0 and γaγb + γbγa = 2ηab. Therefore γ0γ0 = 1 and γ0† = γ0. – 5 – (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) Dirac operator for massless Dirac fields satisfies the identity10 Thus, D = DL + DR where DL := PRDPL, DR := PLDPR. γ5Dγ5 = −D. S = SL + SR, The left-handed and right-handed components of Dirac field decouple consequently, i.e., where, for example, SL := i Z dDyp−g(y) ψL(y)DL(y)ψL(y). Eq. (1.1) implies that the Feynman propagator decomposes accordingly, SFL(x, x′) := PLSF (x, x′)PR, SFR(x, x′) := PRSF (x, x′)PL. In two dimensions, we can choose γ0 = σ1 and γ 5 = σ3 where σi denote the Pauli matrices. Eq. (2.19) implies that SFL(x, x′) has only one nonzero component which we denote by S+(x, x′). Similarly we denote the non-zero component of SFR(x, x′) by S−(x, x′). Following eq. (2.18) we obtain Eq. (2.10) reads, In four dimensions we choose (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) (2.25) SF−(x, x′) ! SF±(x′, x) ⊜ −SF±(x, x′)∗. SF(L)(x, x′) ! 0 , S(L)(x′, x) ⊜ −SF(L)(x, x′)†. F – 6 – SFR(x, x′) by SF(L)(x, x′) and SF(R)(x, x′) respectively. Therefore, Following eqs. (2.19) and (2.20), we denote the nontrivial components of SFL(x, x′) and and eq. (2.10) reads, e.g., 10γ5 anti-commutes with γa and γ5γ5 = 1. PL := 1−γ5 2 and PR := 1+γ5 are projection operator which 2 separate the left-handed and the right-handed components of ψ = ψL + ψR defined by ψL := PLψ and ψR := PRψ. Two dimensional Minkowski spacetime In this section we review the second quantization of massless Dirac fermions in inertial reference frames in two dimensional Minkowski spacetime. We show that one-particle states are imprinted, in a retrievable manner, in the short distance singularity of the Green’s function with Feynman boundary condition. We will use this result to justify the definition of left-mover vacuum state and left-mover one-particle states on a time dependent curved background, later in section 4.3. is γμ∂μψ(x) = 0 where, γ0 := σ1 and γ1 := −iσ2 are Dirac matrices. Introducing In Minkowski spacetime whose line element is ds2 = dx02 − dx12, Dirac field equation correspond to the light-cone coordinates x± := x0 ± x1. left-handed and right-handed components, ψ+ and ψ− respectively, We also introduce γ5 := γ γ 0 1 = σ3 and use it to decompose the Dirac field ψ into its Dirac operator can be given as where satisfying The corresponding field equation is σ± := , γ5ψa = −aψa, a = ±. ∂aψ−a = 0, a = ±, implying that the left-handed component ψ+(x) = ψ+(x+) is left-mover and the righthanded component ψ−(x) = ψ−(x−) is right-mover. The quantized massless Dirac field is given by ψ(x) = ψ−(x) + ψ+(x) and ψ(x) = ψ−(x) + ψ+(x), where p · x := |p| x0 − p x1, ψa(x) := ψa(x) := Z ∞ dp ua(p) −∞ 2π p2 |p| Z ∞ dp u¯a(p) −∞ 2π p2 |p| Apae−ip·x + Bpa†eip·x , Apa†eip·x + Bpae−ip·x , γ5ua(p) = −aua(p), a = ±, – 7 – (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) and u¯a(p) := ua(p)†γ0. Apa and Apa†, and Bpa and Bpa† are the annihilation and creation operators for fermions (A-particles) and anti-fermions (B-particles) with helicity a and momentum p respectively. The field equation reads whose solution is11 (|p| + ap)ua(p) = 0, a = ± , ua(p) = p2 |p|θ(−ap) 1 − aγ5 2 anti-fermions. Similarly, ψ+(x) annihilates left-moving fermions and creates left-moving and |B; p, ai := p2 |p|Bpa† |0i respectively. These can be used to show that The vacuum state |0i is defined to be the state such that Apa |0i = Bpa |0i = 0. The one-particle states of fermions and anti-fermions are defined by |A; p, ai := p2 |p|Apa† |0i ψa(x) = h ψa(x)| = u¯a(p)eip·x |A; p, ai , ua(p)e−ip·x hA; p, a| , SA(x, x′) := given by where ψa(x) := ψa(x) |0i and hψa(x)| := h0| ψa(x). Using the orthogonality of the oneparticle state, hA; p, b|A; p′, ai = 4π |p| δ(p − p′)δab one verifies that the matrix element ψ(x)|ψ(x′) decomposes according to SA = SA− + SA+ whose entries are SrAsa(x, x′) = where ura(p) is the r-th component of ua(p). SAa encodes the amplitude for a particle whose helicity is a propagating from x′ to x. Similarly, – 8 – (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) | ψa(x)i = ψa(x) = ua(p)eip·x |B; p, ai , u¯a(p)e−ip·x hB; p, a| , into its helicity components SB = SB− + SB+ whose entry hB; p, b|B; p′, ai = 4π |p| δ(p − p′)δab decomposes the amplitude SB(x, x′) := where |ψa(x)i := ψa(x) |0i and ψa(x) := h0| ψa(x). Once again, the orthogonality relation ψ(x′)|ψ(x) SrBsa(x, x′) = gives the amplitude for an antiparticle with helicity a to propagate from x to x′. 11We have required that ua(p)†ua(p) = 2 |p| θ(−ap). This normalization is reflected in eq. (3.12) and (3.13). one verifies that Using the equality ua(p)u¯a(p) = 2 |p| θ(−ap)σ−a, SrBsa(x′, x) = SrAsa(x, x′) = 0 Z ∞ dp e−ip(xa−x′a)σr−sa. Z ∞ 0 dp e−ipx = πδ(x) − i dp sin px, 2π in which, δ(x) := ddx θ(x) denotes the Dirac delta function, one obtains 1 2 a δ(xa − x′ ) − 2π 0 i Z ∞ dp sin p(xa − x′a) σr−sa. The Feynman propagator is given by SF rs := θ(x0 − x′0)SrAs(x, x′) − θ(x′0 − x0)SrBs(x, x′). Therefore, SF = SF−(x, x′)σ+ + SF+(x, x′)σ− where, 3.1 Upside down approach SFa (x, x′) = 1 2 sgn(x0 − x′0)δ(xa − x′a) − 2π 0 i Z ∞ dp sin p(xa − x′a) . Now we put things the other way around. We compute a Green’s function G(x, x′) for the Dirac operator γμ∂μ satisfying the condition (2.22), and impose the Feynman boundary condition in order to obtain SF (x, x′). We show that one-particle states and the vacuum state can be recognized in this way. To this aim, we assume that the Green’s function is decomposed into its left-handed and right-handed components G (a)(x, x′) according to eq. (2.21), i.e., Consequently, the field equation (3.2) reads whose solution is also satisfying eq. (2.22). G(x, x′) = X G (a)(x, x′)σ−a. a=± 2∂−aG(a)(x, x′) = δ2(x − x′), a = ± , G (a)(x, x′) ⊜ 1 2 sgn(x0 − x′0)δ(xa − x′a), The expression (3.26) reproduces the first term on the right hand side of eq. (3.23). Recalling that for x 0 > x′0 and x 0 < x′0, the Feynman propagator corresponds to the amplitude for particles with positive frequency to propagate from x′ to x and from x to x′ respectively, the second term on the right hand side of equation (3.23), which is a solution (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) (3.26) HJEP07(218) to the homogeneous field equation, is uniquely determined. In fact eq. (3.20) shows that this term removes the negative frequencies in the spectrum of δ(xa By rewriting the result in terms of the θ function similarly to eq. (3.22) we recognize SAa and SBa as given by eq. (3.19). σa can be factorized identically to eq. (3.18) whose solution ua(p), given in eq. (3.11), is unique up to a phase factor. As a result, the matrix elements SrAsa(x, x′) and SrBsa(x, x′) can be recognized as given by eq. (3.14) and eq. (3.17) respectively. Finally, they can be factorized according to eqs. (3.12) and (3.13) and eqs. (3.15) and (3.16) respectively. At this point we postulate the orthogonal one-particle states |A; p, ai and − x′a). |B; p, ai and subsequently postulate the vacuum state |0i. 4 Two dimensional curved spacetime We denote the spacetime metric by gμν and the Minkowski metric by12 HJEP07(218) The local frame is identified by the tetrad eμ(a) and its inverse gμν = eμ(+)eν (−) + eμ(−)eν (+), where eμ(±) := √12 (eμ(0) ± eμ( 1 )). We assume that gμν is a continuous function of the spacetime coordinates, g := det gμν < 0 and g11 < 0. That is to say, the spacetime is orientable and dx0 = 0 corresponds to space-like intervals. Thus g00 > 0 and we choose the tetrad such that e0(a) > 0. Furthermore, we select the ± sign such that (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) satisfying the identity we choose the inertial frame where, gμν is the inverse of gμν . We use the Einstein summation notation when we sum over spacetime indices µ, ν = 0, 1. In the flat spacetime limit, where and consequently As a result where Eμ(a) := √ −g eμ(a). e0(+)e1(−) > e0(−)e1(+), √ −g = e0(+)e1(−) − e0(−)e1(+). E0(a) = ae1(−a), E1(a) = −ae0(−a), 12We are describing the Minkowski spacetime in the light-cone gauge. We also recall that the classical trajectory of massless particles are light-like. Assuming that tively, eq. (4.9) reads where ψ = ψ − ψ+ ! , S = S+ + S−, i Z 2 Z Sa := √ d2x Eμ(a)(x) ψa(x)∗ ←∂→μ ψa(x) = i d2x p−g(x)ψa(x)∗D−aψa(x), The action is given by [13] where, and the operator ←→∂ is defined according to the rule where ψ+ and ψ − are the so-called left-handed spinor and the right-handed spinor respecϕ1(x) ←∂→μ ϕ2(x) := −(∂μϕ1(x))ϕ2(x) + ϕ1∂μϕ2(x). in which, we have integrated by part and dropped a boundary term to obtain the second equality, and D−a(x) := s −2 g(x) Eμ(a)(x)∂μ + 1 2 ∂μEμ(a)(x) , is the Dirac operator in the corresponding sector. In the flat spacetime limit, and in the inertial frame (4.5) This theory is invariant under local ‘Lorentz’ transformations13 Da = 2∂a, a = ± . eμ(a)(x) → ψa(x) → (λ)eμ(a) (x) = eaλ(x)eμ(a)(x), (λ)ψa(x) = ea λ(2x) ψa(x). ψa(x) := u(a) z(a)(x) e−R(a)(x), The solution to the classical field equation D−a(x)ψa(x) = 0 is given by 13Recall that we are using the light-cone coordinates. The fastest way to recognize the generator of Lorentz Thus for spacetime coordinates, i.e., in the real spin 1 representation, it is given by σ3. transformation in this representation is to note that for spinors, the generator is given by 4i [γ0, γ1] = 2i σ3. in which u(a) is a smooth function, and z(a) and R (a) are real-valued functions. R (a) solves Eμ(a)(x)∂μR (a)(x) = 1 2 ∂μEμ(a)(x), which according to the Peano existence theorem has at least one solution only if Eμ(a)(x) are continuous [14]. z(a) are given by the following equations ∂0z(a) = −aE1(a)e−2R(a) , ∂1z(a) = aE0(a)e−2R(a) . Eμ(a)(x)∂μz(a)(x) = 0. ∂μz(a) = eμ(−a)e−2R(a) , µ = 0, 1, ds2 = 2e2R(+) e2R(−) dz(+)dz(−). Eq. (4.21) implies that [∂0, ∂1]z(a) = 0, which is the necessary condition for the existence of z(a), and HJEP07(218) Using eq. (4.8), one can rewrite eqs. (4.22) and (4.23) in the following way, which together with eq. (4.1) gives the celebrated result that two dimensional manifolds are conformally flat, 4.1 Feynman propagator and simultaneously eq. (2.22). Define Eq. (4.24) and eq. (4.21) give D−a(x) δT(a)(x, x′)e−R(a)(x) = and eq. (4.23) gives, In section 2 we observed that the Feynman propagator SFa (x, x′) solves the equation D−a(x)SFa (x, x′) = δ2(x − x′), a = ± , 1 p−g(x) δT(a)(x, x′) := 2− 23 sgn(x0 − x′0)δ z(a)(x) − z(a)(x′) . = (−g(x))− 12 E0(a)(x′)δ(x0 − x′0)δ z(a)(x) − z(a)(x′) e−R(a)(x), δ(x0 − x′0) δ z(a)(x) − z(a)(x′) = δ(x0 − x′ ) E0(a)(x′) 0 δ(x1 − x′1) e2R(a)(x′), where we have noticed that E0(a) > 0. Therefore SFa x, x′ ⊜ δT(a)(x, x′)e−R(a)(x)−R(a)(x′), Since z(a)(x, x′) = 0 indicates a light-like curve, δT(a)(x, x′) in eq. (4.31) can be interpreted in terms of the amplitude corresponding to the propagation of light-like modes similarly to the flat spacetime discussed in section 3.1. In this way, the factor exp(−R should be considered as an x-dependent normalization, which obstructs an interpretation (a)(x)) of the amplitude in terms of one-particle states. Solutions to the homogeneous field equation do not cancel out nonzero R a reflection of the spacetime curvature. (a)(x). In the next subsection we show that this effect is (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) (4.27) (4.28) (4.29) (4.30) (4.31) In this subsection, we study the spin connection in D-dimensional spacetimes. For simplicity, here and also later in sections 5 and 6, we use the Einstein summation notation when we sum over frame indices a, b, c, d = 0, · · · , D. The Dirac operator is given by where denotes the spin connection. Here, D(e) := eμ(a)γa∂μ + Ω, i μ Ω := − 2 e (a)γaΩμ(e), Ωμ(e) := eν (a)∇μeν (b)Σab, ∇μ denotes the Levi-Civita connection, and Σab := 4i [γa, γb] satisfying the Lorentz algebra i[Σab, Σcd] = ηacΣbd + ηbdΣac − (a ↔ b). Under a local Lorentz transformation given by U (ξ) := exp i − 2 ξabΣab , i we have U (ξ)−1γaU (ξ) = Λ(ξ)abγb, where Λ(ξ) := exp − 2 ξabJ ab . J ab, whose entries are given by [J ab]cd = i(ηacδdb − ηbcδda), satisfy the Lorentz algebra similarly to Σab. One can show that Ωμ((ξ)e) = U (ξ)ΩμU (ξ)−1 + Ξ(ξ)μ, where Ξ(ξ)μ := 2iU (ξ)∂μU (ξ)−1 and Ω12 = ∇μeμ(−), Ω21 = ∇μeμ(+). (4.32) (4.33) (4.34) (4.35) (4.36) (4.37) (4.38) (4.39) (4.40) (4.41) (4.42) (4.43) (ξ)eμ(a) := Λ(ξ)abeμ(b). D((ξ)e) = U (ξ)D(e)U (ξ)−1. ℧μν (Ω) := ∇μΩν − ∇ν Ωμ − 2 i Since ℧μν (Ξ(ξ)) = 0, and ℧μν (Ω(e)) = Rμνρσeρ(a)eσ(b)Σab, in which Rμνρσ denotes the Riemann tensor, we conclude that in a curved spacetime, Ω can not be eliminated by local Lorentz transformations. In two dimensions, R 2 Rμνρσ = (gμρgνσ − gμσgνρ), where R is the Ricci scalar. Consequently, ℧01(Ω(e)) = iR√ by 2 anti-diagonal matrix whose non-zero entries are −gσ3. Furthermore, Ω is a 2 Henceforth, we denote it by Noting that one verifies that D(e) = [[D+, D−]]/. Left-handed frames Ω = [[∇μeμ(−), ∇μeμ(+)]]/. ∂μEμ(a) = √ −g∇μeμ(a), Eq. (4.45) implies that under a general coordinate transformation, ∂μEμa(x) → ∂μ′E′μ(a)(x′) = det ∂μEμ(a)(x), ∂x ∂x′ in which the determinant is the Jacobian of the coordinate transformation. Eq. (4.18) gives the local Lorentz transformation rule, ∂μEμ(a)(x) → ∂μ(λ)Eμ(a)(x) where, eaλ(x)∂μ(λ)Eμ(a)(x) = ∂μEμ(a)(x) − aEμ(a)(x)∂μλ(x). The local Lorentz transformation can be used to satisfy the left-mover gauge ∂μEμ(+)(x) = 0. The transformation rule (4.46) implies that such left-mover frames are independent of the coordinate system. Explicitly, if we give the metric in the conformal gauge, ds2 = e2ωdx+dx−, the left-handed frame is described by R (+)(x) = 0 and R(−)(x) = ω(x). Using (4.49) in eq. (4.14) gives e−(+) = √2e−2ω, e−(−) = 0, e+(+) = 0, e+(−) = √2, S+ = i Z d2x ψ+(x)∗←∂→−ψ+(x), and consequently, eq. (1.1) implies that the Feynamnn propagator is given by 1 2 similarly to eq. (3.23). Equivalently, using (4.49) in eq. (4.16) we obtain D− = 2e−2ω∂ − = e−2ω (∂0 − ∂1) , and eq. (4.52) solves eq. (4.27) and eq. (2.22). Following section 3.1, we can interpret SF+(x, x′) as the propagation amplitude of fermionic one-particle states. Furthermore, following the standard approach to the conformal field theory in two dimensions, one can identify the left-movers in the left-mover gauge as a chiral c = 12 conformal field theory [15, 16]. Eq. (4.47) implies that the chiral symmetry is generated by λ(+) satisfying the condition Eμ(+)∂μλ(+) = 0, i.e., λ(+) = λ(+)(x+) [17]. This theory is the typical example of systems with gravitational anomaly [13, 18–20]. (4.44) (4.45) (4.46) (4.47) (4.48) (4.49) (4.50) (4.51) (4.52) (4.53) HJEP07(218) Four dimensional curved spacetime Following [9] we denote the Minkowski metric by η = diag( 1, −1, −1, −1 ), and use Dirac gamma matrices γa := [[σa, σ¯a]]/ and γ5 := [[−1, 1]]\. The notation we are using in writing γ5 indicates that it is block-diagonal. In writing γa we have considered them as 2 by 2 anti-diagonal matrices whose entries are the 2 by 2 matrices σ0 := σ¯0 := 1 and σ¯i := − σi. In this notation, eq. (4.34) reads Ωμ = [[ΩLμ, ΩRμ]]\, in which ΩRμ := − σ2Ω∗Lμσ2. Furthermore, eq. (4.36) reads where In section 4.2 we showed that in a curved spacetime, U (ξ) = [[UL(ξ), UR(ξ)]]\, UR(ξ) = σ2UL(ξ)∗σ2. Ωμ 6= 2iU (ξ)−1∂μU (ξ), so, eqs. (5.1) and (5.2) imply that in four dimensional curved backgrounds, there is no local Lorentz transformation Λ(ξ) such that ΩLμ(ξe) = 0. In other words, the curvature effect can not be removed from the left-handed sector by means of local Lorentz transformations. Eq. (4.33) gives Ω = [[ΩR, ΩL]]/, where ΩR := σ2Ω∗Lσ2, ΩL := ζ(a)σ¯a, ζ(a) := 1 2 ∇μeμ(a) + i μ 4 e (a)Iμ, Iμ := ǫabcdeμ(a)eρ(b)eν (c)∂ρeν (d), 1 1 a and ǫabcd := ηaeǫebcd in which ǫabcd is the totally antisymmetric tensor so that ǫ0123 = 1. Iμ(x) is frame dependent. That is, for a local Lorentz transformation Λ(ξ), eq. (4.38) gives ξ Iμ = Iμ + ǫabcdηcneμ(a)eρ(b)Λpd∂ρΛpn, so we can choose a frame in which Iμ = 0. In the normal neighborhood of x′ = 0 [2], gμν (x) = ημν − 3 Rμανβxαxβ − 6 ∇γ Rμανβxαxβxγ + O(x4), this frame is given by Iμ(x) = O(x3). eμa(x) = δμa − 6 R αμβxαxβ − 12 ∇γ Raαμβxαxβxγ + O(x4) , as can be verified by using the Bianchi identities for the Riemann tensor to show that Eq. (4.32) reads D(e) = [[DR, DL]]/ in which DR := σ2D∗Lσ2 and DL := σ¯aeμ(a)∂μ + ΩL = σ¯a eμ(a)∂μ + ζ(a) , 1 1 (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) where we have used eq. (5.5). The Dirac field equation D(e)Ψ = 0 gives DLψL = 0, DRψR = 0, projection operators PL = 1−γ5 and PR = 1+2γ5 . 2 where ψL and ψR are the left-handed and right-handed components of Ψ separated by the Following the argument in section 2, eqs. (5.21) and (5.22) show that This result can be interpreted as the propagation amplitude of one-particle states with an x-dependent normalization because and choose the tetrad, Eq. (5.6) gives and eq. (5.11) gives ds2 = e2ω(x)ημν dxμdxν , eμ(a)(x) = eω(x)δμa. ζ(a) = 2 3 e−ω(x)δμa∂μω(x), DL(x) = σ¯aδμae−ω(x) ∂μ + 3 2 ∂μω(x) . Consider the two-point function where and xa := δμa xμ. Using the identity we find that Furthermore, Gc(fL)(x, x′) := i Z d4p p · σ DL(x)Gc(fL)(x, x′) = δ4(x − x′) p−g(x) . Gc(fL)(x′, x) ⊜ −Gc(fL)(x, x′)†. SF(L)(x, x′) ⊜ Gc(fL)(x, x′). Gc(fL)(x, x′) = G1(Lp)(x, x′)e− 23 (ω(x)+ω(x′)), where G1(Lp)(x, x′) := i Z gives the Feynman propagator in an inertial frame in Minkowski spacetime.14 Such an interpretation is supported by the classical symmetry of massless Dirac fields in four dimensions under the conformal map The left handed-field ψL can be further decomposed into its spin-up and spin-down components by means of the projection operators which separate the spin-up component and the spin-down component of the field along the third direction respectively. That is to say, ψL = ψL↑ + ψL↓, where (5.25) (5.26) (5.27) (5.28) (5.29) (5.30) (5.31) (5.32) (5.33) (5.34) and ψL↓ := pˆ↓ψL. Assuming that ψL↓ = 0, the field equation (5.12) gives DLψL↑ = 0 and consequently Eq. (5.29) can be considered as the classical field equation corresponding to the action, S[ψL↑†, ψL↑] = i Z d4xp−g(x)ψL↑†DLψL↑. F The corresponding Feynman propagator S↑(L)(x, x′) can be computed by the path integral (1.1). Eq. (1.1) and eq. (5.28) already imply that S↑(L)(x, x′) has only one nonzero component which we denote by S↑(L)(x, x′). More explicitly, F F S↑(L)(x, x′) = pˆ S↑(L)(x, x′)pˆ↑ = S↑(L)(x, x′)pˆ↑. F ↑ F F Similarly we use the symbol DL↑ to denote the nonzero component of pˆ↑DLpˆ↑. Following section 2 we know that the Feynman propagator SF↑(L)(x, x′) satisfies the following equations (e2ωημν , ψ) → (ημν , e 32 ωψ). pˆ↑ ≡ 1 + σ 2 3 , pˆ↓ ≡ 1 − σ 3 2 , DL↑SF↑(L)(x, x′) = (−g(x))− 12 δ4(x − x′), S↑(L)(x′, x) ⊜ −SF↑(L)(x, x′)∗. F DL↑ = 2e−ω(x) ∂− + 3 2 ∂−ω(x) , Eq. (5.17) implies that in the frame of reference (5.15) replacing p/ by p · σ therein. 14G1(Lp)(x, x′) is given by the right hand side of eq. (1.3) after setting m = 0 in eqs. (1.16) and (1.17) and where x± := x0 ± x3. Consequently in which S↑(L)(x, x′) ⊜ SF+(x, x′)δ2(x⊥ − x′⊥)e− 32 (ω(x)+ω(x′)), F δ2(x⊥ − x′⊥) := δ(x1 − x′1)δ(x2 − x′2), and SF+(x, x′) is given in eq. (3.23). Eq. (5.35) implies that S↑(L)(x, x′) corresponds to F propagating modes confined to a two dimensional subspace. Since SF+(x, x′) is the Feynman propagator of left-moving spinors in a two dimensional Minkowski spacetime we conclude that similarly to the standard result (5.24), the Feynman propagator (5.35) can be interpreted in terms of one particle states with an x-dependent normalization. So we sacrificed the spin-down component of the field but gained nothing new in the frame of reference (5.15). Now consider another frame of reference given by eμ(+)dxμ = eμ(−)dxμ = e4ωdx−, eμ(a)dxμ = eωdxa, 2 1 e−2ωdx+, a = 1, 2, (5.35) (5.36) (5.37) (5.38) (5.39) (5.40) (5.41) (5.42) (5.43) This implies that the Minkowski metric in the local frame is given in the light-cone gauge η−+ = 1, η±a = 0, ηab = −δab, for a, b = 1, 2. Eq. (5.6) gives ζ(−) = 0,15 thus and eqs. (5.32) and (5.33) give DL↑ = e−4ω (∂0 − ∂3) , S↑(L)(x, x′) = SF+(x, x′)δ2(x⊥ − x′ ). F ⊥ This can be also verified by using the path integral (1.1) and noting that in this frame of reference, eq. (5.30) reads Z S[ψL↑†, ψL↑] = 2i F Following the argument in section 3.1, S↑(L)(x, x′) in eq. (5.42) can be interpreted as the propagation amplitude of one-particle states localized on a two dimensional Minkowski spacetime. is proportional to ∂μ(√ 15Both of eqs. (5.37) and (5.38) give √−ge−(−) = 1 and eμ(−) = 0 for µ = +, 1, 2. Thus, Re ζ(−) which −geμ(−)) is zero. The imaginary part of ζ(−) is given by ǫ−bcdeμ(b)eν(c)∇μeν(d). Eq. (5.39) implies that the contribution from the b = − terms in Im ζ(−) is zero. Now consider the contribution from the c = − terms. Eqs. (5.37) and (5.38) imply that only d = + contributes in eν(−)∇μeν(d). Therefore the c = − terms (and similarly the d = − terms) add zero to Im ζ(−). Let ψL be the left-handed component of a Dirac field. Suppose that we project ψL into its “spin-up” component where pˆ↑ is a projection operator whose Lorentz transformation is given by Therefore, the Lorentz transformation ξψL = UL(ξ)ψL induces a similar transformation ξψL↑ = UL(ξ)ψL↑. Consider the operator In summary, in the model (5.30) for the separated spin-up component of the left-handed Dirac field, the Feynman propagator can be described in terms of propagating modes localized on a two dimensional subspace and there exists a frame of reference in which the one-particle states can be defined similarly to the second quantization in two dimensional Minkowski spacetime. In fact, in this model, the frame given by eqs. (5.37), (5.38) and (5.39) is reminiscent of the left-handed frame in two dimensions. In the next section we show that the action (5.30) enjoys local Lorentz symmetry in addition to the diffeomorphism invariance and such frames exist in general. Consequently the spin-up one-particle states and the corresponding vacuum state are well-defined in curved spacetimes. 6 One particle states HJEP07(218) S[ψL↑†, ψL↑] := i Z d x 4 √ −gψL↑†DL↑ψL↑. DL↑ψL↑ = 0. F S↑(L)(x, x′) = pˆ S↑(L)(x, x′)pˆ† , ↑ F ↑ DL↑S↑F(L)(x, x′) = 1 p−g(x′) δ4(x − x′)pˆ† , ↑ S↑(L)(x′, x) ⊜ −S↑F(L)(x, x′)†. F Eq. (4.39) implies that the Lorentz transformation maps DL to Eq. (5.3) implies that UR(ξ)† = UL(ξ)−1, and consequently ψL↑†DL↑ψL↑ is invariant under local Lorentz transformations. Therefore, local Lorentz transformation are symmetries of the action The classical field equation reads The path-integral (1.1) and definition (6.1) imply that the corresponding Feynman propagator, which we denote by S↑(L)(x, x′), satisfies the identity F and equations, (6.1) (6.2) (6.3) (6.4) (6.5) (6.6) (6.7) (6.8) (6.9) The spin-up component of a left-handed fermion along the x3-direction is separated by In flat spacetime, using eq. (6.10) and eq. (5.11) in eq. (6.3) we obtain pˆ↑ ≡ 1 + σ 2 3 . DL↑ = pˆ↑(∂0 − ∂3). ψL↑ = e−ipx+ eiq⊥·x⊥ p > 0, Thus the plane-wave solution of the classical field equation (6.6) is given by up to a normalization constant, where and q ⊥ ∈ R2. Following eqs. (6.8) and (6.9) we obtain S↑(L) = SF+(x, x′)δ2(x⊥ − x′⊥)pˆ↑, F gμν = (−g)− 12 eμ(+)g−ν + eν(+)g−μ − X eμ(a)eν(a). a = −, 1, 2. 2 a=1 (6.10) (6.11) (6.12) (6.13) (6.14) (6.15) (6.16) (6.17) (6.18) (6.19) (6.20) (6.21) (6.22) (6.23) where SF+(x, x′) is given in eq. (3.23). So, the four dimensional Feynman propagator is given by the amplitude of left-moving spinors propagating in a two dimensional subspace. Now consider a curved spacetime equipped with coordinates x± and x ⊥ such that g−− = 0. Suppose that the Minkowski metric in local frames is given in the light cone gauge (5.40). Choose a local frame in which ∂(−) = (−g)− 12 ∂−, i.e., e−(−) = (−g)− 12 , eμ(−) = 0, µ = +, 1, 2. ηab = eμ(a)eμ(b), a, b = ±, 1, 2, The identity implies that These equations also show that for µ, ν = +, 1, 2, e−(+) = (−g) 2 , 1 e−(a) = 0, g−μ = (−g) 2 eμ(−), 1 It is easy to verify that eμ(a) satisfy eqs. (6.20)–(6.23) and also eq. (6.24). Using eqs. (6.17) and (6.18) in eq. (5.6) one verifies that Re ζ(−) = 1 2p−g(x) ∂μ = 0. Also, by using eqs. (6.17), (6.18), (6.21) and (6.24) in eq. (5.7) one can show that Im ζ(−) = 0.16 Thus using eq. (6.10) in eq. (6.3) to separate the spin-up component of DL (eq. (5.11)) along the third direction, we obtain Eqs. (6.8) and (6.9) imply that similarly to the flat spacetime, the Feynamn propagator is given by eq. (6.15). In brief, after using eq. (6.28) in eq. (6.5) we obtain DL↑ = 2pˆ↑(−g)− 12 ∂−. Z S[ψL↑†, ψL↑] := 2i d4xψL↑†∂ ψ↑ , − L eν1 eν2 cos ϕ − sin ϕ ! sin ϕ cos ϕ e′ν1 ! e′ν2 , ν ∂−ϕ = e′ ( 1 )∇−e′ν( 2 ). (6.24) (6.25) (6.26) (6.27) (6.28) (6.29) (6.30) (6.31) We also assume that where ∇μ denotes the Levi-Civita connection. This requirement can be satisfied by using local rotations in the (( 1 ) − ( 2 )) plane. To see this, start with some tetrad e′ν(a) satisfying eqs. (6.20)–(6.23) and define eν(±) := e′ν(±) and in which ϕ solves the equation and the path integral (1.1) results in eq. (6.15). Following section 3.1, the corresponding one-particle states and vacuum state can be postulated similarly to the second quantization in Minkowski spacetime. 6.1 The Kerr solution The x0 ordering in eq. (1.28) is a “time” ordering only if the vector ∂0 is timelike which is not the case inside an ergosphere. Since the Feynman propagator (1.28) can be also understood as an x− ordered expression 16The argument is similar to footnote 15. As an example, consider the Kerr solution whose line element in the Kerr coordinates is given by ds2 = −2dr du+ − a sin θ2dφ+ −ρ2dθ2 − ρ−2 sin θ2 (r2 + a2)2 − Δa2 sin θ2 dφ+2 +4amρ−2r sin θ2dφ+du+ + (1 − 2mrρ−2)du+2. where ρ2 := r2 + a2 cos θ2, Δ := r2 + a2 − 2mr, and m and ma are constants representing the mass and the angular momentum as measured from infinity [21]. Since grr = 0, ∂r is a null vector and we can identify x− with r and choose any suitable function of the other coordinates as x+ = x+(u+, θ, φ+). In this way, the Feynman propagator (6.31) is r-ordered. A more familiar description can be obtained by solving (r2 + a2)Δ−1dr = 1 2 (du+ − du−), for r and inserting the function r = r(u−, u+) in eq. (6.32) to obtain the line element in the (u±, θ, φ+) coordinates. In these coordinates gu−u− = 0 and we identify x− with u−. Noting that the Kerr coordinates in terms of the Boyer and Lindquist coordinates (t, r, θ, φ) (6.32) (6.33) (6.34) (6.35) (6.36) (7.1) (7.2) (7.3) ∂(−) = (−g)− 21 ∂−, eν ( 1 )∇−eν( 2 ) = 0, ψL↑ := 1 + σ3 2 ψL, where ∇μ denotes the Levi-Civita connection, g is the determinant of the spacetime metric in which, σ3 is the third Pauli matrix and ψL is a left-handed massless Dirac field. are given by and consequently 7 Conclusion du± = dt ± (r2 + a2)Δ−1dr, dφ+ = dφ + aΔ−1dr , u − = t − r + 2m ln r + O(r−1), one verifies that the u − ordering of the Feynman propagator reproduces the ordinary t ordering via eq. (6.30) asymptotically. tetrad eμ(a), A spinor field in curved background is defined by means of local Lorentz transformations. We have shown that in a four dimensional curved background, in general, there exists a spinor field ψL↑ which is annihilated by a null vector field ∂− in a certain frame of reference. In a coordinate system given by x± := x0 ± x3, and x ⊥ := (x1, x2) such that the metric component g−− = 0, this frame of reference is identified by the following conditions on the The corresponding Feynman propagator is given by S↑(L) = SF+(x, x′)δ2(x⊥ − x′ ), F ⊥ (7.4) F in which SF+(x, x′) denotes the Feynman propagator obtained by means of the second quantization of a left-moving massless Dirac field in two dimensional Minkowski spacetime. Therefore S↑(L) can be interpreted in terms of propagating one-particle states confined to a two dimensional Minkowski spacetime equipped with coordinates x±, and the corresponding vacuum state is well-defined similarly to the second quantization in Minkowski left-handed massless Dirac field travelling along the x3-axis. spacetime. In the flat spacetime limit, ψL↑ is reminiscent of the spin-up component of a This line of thought is motivated by an observation in two dimensions. As we have argued in detail, in a two dimensional curved background there exists, in general, a local frame in which the left-moving massless Dirac field is annihilated by a null vector field ∂ , and consequently, the corresponding Feynman propagator equals SF+(x, x′). Therefore the Feynman propagator can be interpreted in terms of propagating one-particle states similarly − to the second quantization in Minkowski spacetime. In such local frames, the curvature effect is totally transmitted to the right-moving sector. Consequently, the interpretation of the corresponding Feynman propagator in terms of propagating right-moving modes requires an x-dependent normalization of the one-particle states. In four dimensions, the chirality is reversed by CPT transformation [13], hence, both of the left-handed and the right-handed components of the massless Dirac field are equally affected by the spacetime curvature. So we have focused on the left-handed sector and separated its spin-up and spin-down components covariantly with respect to the local Lorentz transformations. We have introduced an action for the spin-up component which enjoys diffeomorphism invariance and local Lorentz transformation. It is given by pˆ↑ → UL(Λ)pˆ↑UL(Λ)−1, Z S[ψL↑†, ψL↑] := 2i S[ψL↑†, ψL↑] := i Z d x 4 √ −gψL↑†DLψL↑, where DL is the Dirac operator for massless fermions in the left-handed sector, and ψL↑ is the spin-up component of the left-handed Dirac field ψL ψL↑ := pˆ↑ψL, in which pˆ↑ is the corresponding projection operator. We have supposed that ψL↑ is in the spin 12 representation of the local Lorentz transformations similarly to ψL. This can be done by considering pˆ↑ as a tensor field. Explicitly, if UL(Λ) denotes the operator corresponding to a local Lorentz transformation Λ in the left-handed sector such that ψL → UL(Λ)ψL, we require that pˆ↑ equals 1+σ3 2 and accordingly. 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Farhang Loran. Fermionic one-particle states in curved spacetimes, Journal of High Energy Physics, 2018, 71, DOI: 10.1007/JHEP07(2018)071