Fermionic one-particle states in curved spacetimes

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 $\partial_-$ 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.


Introduction
The study of massless Dirac fermions on non-stationary curved backgrounds has diverse theoretical and observational motivations [1,2,3,4,5,6]. Al-though 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 where ψ and ψ are Grassmann fields, S denotes the action and Eq.(1.1) implies that S F (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. In four dimensional Minkowski spacetime, Dirac operator in an inertial (non-accelerating) reference frame is (i∂ /−m), 1 and S F (x, x ′ ) equals the Feynman propagator 2 (1.3) in which, the amplitudes represent the propagation of positive-energy particles from x ′ to x and from 1 m denotes the mass of Dirac particle, ∂ / := γ a ∂ a and γ a are Dirac matrices. 2 θ denotes the Heaviside step function. θ(x) equals 1 and 0, for x > 0 and x < 0 respectively.
(1. 15) Consequently, (1.17) The spinors anti-commute at spacelike separation, i.e., This in turn indicates that the particles represented by Dirac field obey Fermi statistics [9]. That is, if we postulate a vacuum state |0 invariant under translation and Lorentz transformation, and ladder operators a r p and b r p such that In general, the concept of positive energy is lost in non-stationary curved spacetime and also in non-inertial reference frames in flat spacetime 5 [10]. As 4 δ D denotes the Dirac delta-function in D-dimensions. The Kronecker delta δ rs equals 1 and 0 for r = s and r = s respectively. 5 A counterexample is the non-inertial frame corresponding to Rindler observers in Minkowski spacetime because the Rindler wedge of Minkowski spacetime is globally hyperbolic [11].
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 time-ordering which has been used in Eq.(1.3), although the Feynman propagator S F (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 (1.23) such that g −− = 0. In a frame of reference given by 7 we separate the spin-up component of the left-handed massless Dirac field along the third direction and compute the corresponding Feynman propagator S where S + F (x, x ′ ) denotes the Feynman propagator of left-moving massless fermions in an inertial frame in a two dimensional Minkowski spacetime 6 There 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. e µ (a) is the tetrad identifying the frame of reference. ∂ (a) := e µ (a) ∂ µ .
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 ⊥ plane indeed. The sign function in Eq.(1.28) can be equivalently given in terms of (1.29) 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 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.

General properties of Feynman propagator
Our starting point is Eq.(1.1) in which where D(y) denotes the Dirac operator. This equation implies that .
Noting that the action S is real-valued, Eq.(2.1) also gives 9 where we have used the complex conjugation rule for Grassmann numbers . (2.5) Eq.(2.2) and Eq.(1.1) together give 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 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

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.
In Minkowski spacetime whose line element is ds 2 = dx 0 2 − dx 1 2 , Dirac field equation is γ µ ∂ µ ψ(x) = 0 where, γ 0 := σ 1 and γ 1 := −iσ 2 are Dirac matrices. Introducing Dirac operator can be given as correspond to the light-cone coordinates x ± := x 0 ± x 1 . We also introduce γ 5 := γ 0 γ 1 = σ 3 and use it to decompose the Dirac field ψ into its left-handed and right-handed components, ψ + and ψ − respectively, The corresponding field equation is The quantized massless Dirac field is given by The vacuum state |0 is defined to be the state such that A a p |0 = B a p |0 = 0. The one-particle states of fermions and anti-fermions are defined by |A; p, a := 2 |p|A a p † |0 and |B; p, a := 2 |p|B a p † |0 respectively.
These can be used to show that where ψ a (x) := ψ a (x) |0 and ψ a (x)| := 0| ψ a (x). Using the orthogonality of the one-particle state, A; p, b|A; p ′ , a = 4π |p| δ(p − p ′ )δ ab one verifies that the matrix element S A (x, x ′ ) := ψ(x)|ψ(x ′ ) decomposes according to S A = S A− + S A+ whose entries are given by where u a r (p) is the r-th component of u a (p). S Aa encodes the amplitude for a particle whose helicity is a propagating from x ′ to x. Similarly, gives the amplitude for an antiparticle with helicity a to propagate from Using the equality The Feynman propagator is given by

Upside down approach
Now we put things the other way around. We compute a Green's function G(x, x ′ ) for the Dirac operator γ µ ∂ µ satisfying the condition (2.22), and impose the Feynman boundary condition in order to obtain S F (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 Consequently, the field equation (3.2) reads also satisfying Eq.(2.22). 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 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(x a − x ′ a ).
By rewriting the result in terms of the θ function similarly to Eq.(3.22) we recognize S Aa and S Ba as given by Eq. (3.19). σ a can be factorized identically to Eq.(3.18) whose solution u a (p), given in Eq. (3.11), is unique up to a phase factor. As a result, the matrix elements S Aa rs (x, x ′ ) and S Ba rs (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, a and |B; p, a and subsequently postulate the vacuum state |0 .

Two dimensional curved spacetime
We denote the spacetime metric by g µν and the Minkowski metric by 12 The local frame is identified by the tetrad e µ (a) and its inverse satisfying the identity where, g µν is the inverse of g µν . We use the Einstein summation notation when we sum over spacetime indices µ, ν = 0, 1. In the flat spacetime limit, where we choose the inertial frame 12 We are describing the Minkowski spacetime in the light-cone gauge. We also recall that the classical trajectory of massless particles are light-like.
where e µ (±) := 1 √ 2 (e µ (0) ± e µ (1) ). We assume that g µν is a continuous function of the spacetime coordinates, g := det g µν < 0 and g 11 < 0. That is to say, the spacetime is orientable and dx 0 = 0 corresponds to space-like intervals. Thus g 00 > 0 and we choose the tetrad such that e 0 (a) > 0. Furthermore, we select the ± sign such that As a result where E µ (a) := √ −g e µ (a) . The action is given by [13] where, and the operator ← → ∂ is defined according to the rule Assuming that where ψ + and ψ − are the so-called left-handed spinor and the right-handed spinor respectively, Eq.(4.9) reads where in which, we have integrated by part and dropped a boundary term to obtain the second equality, and is the Dirac operator in the corresponding sector. In the flat spacetime limit, and in the inertial frame (4.5) The solution to the classical field equation D −a (x)ψ a (x) = 0 is given by 20) in which u (a) is a smooth function, and z (a) and R (a) are real-valued functions.
13 Recall that we are using the light-cone coordinates. The fastest way to recognize the generator of Lorentz transformation in this representation is to note that for spinors, the generator is given by i 4 [γ 0 , γ 1 ] = i 2 σ 3 . Thus for spacetime coordinates, i.e., in the real spin 1 representation, it is given by σ 3 .
which according to the Peano existence theorem has at least one solution only if E µ (a) (x) are continuous [14]. z (a) are given by the following equations

Feynman propagator
In section 2 we observed that the Feynman propagator S a F (x, x ′ ) solves the equation Eq.(4.24) and Eq.(4.21) give and Eq.(4.23) gives, where we have noticed that E 0 (a) > 0. Therefore, T (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 (a) (x)) should be considered as an x-dependent normalization, which obstructs an interpretation of the amplitude in terms of one-particle states. Solutions to the homogeneous field equation do not cancel out nonzero R (a) (x). In the next subsection we show that this effect is a reflection of the spacetime curvature.

Curvature
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.
Dirac operator is given by where  Under a local Lorentz transformation given by , satisfy the Lorentz algebra similarly to Σ ab . One can show that Consequently, Since ℧ µν (Ξ(ξ)) = 0, and ℧ µν (Ω(e)) = R µνρσ e ρ (a) e σ (b) Σ ab , in which R µνρσ denotes the Riemann tensor, we conclude that in curved spacetime, Ω can not be eliminated by local Lorentz transformations.

Left-handed frames
Eq.(4.45) implies that under a general coordinate transformation, in which the determinant is the Jacobian of the coordinate transformation. Eq.(4.18) gives the local Lorentz transformation rule, The local Lorentz transformation can be used to satisfy the left-mover gauge ∂ µ E µ (+) (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, gives and consequently, Eq.(1.1) implies that the Feynamnn propagator is given by and Eq.(4.52) solves Eq.(4.27) and Eq.(2.22). Following section 3.1, we can interpret S + F (x, x ′ ) as the propagation amplitude of fermionic one-particle states.
In section 4.2 we showed that in a curved spacetime, and ǫ a bcd := η ae ǫ ebcd in which ǫ abcd is the totally antisymmetric tensor so that ǫ 0123 = 1. I µ (x) is frame dependent. That is, for a local Lorentz transformation Λ(ξ), Eq.(4.38) gives so we can choose a frame in which I µ = 0. In the normal neighborhood of (5.9) this frame is given by

Conformally flat spacetimes
Suppose  16) and Eq.(5.11) gives Consider the two-point function where p · q := p a q a , (5.19) and x a := δ µ a x µ . Using the identity σ · pσ · p = p · p, (5.20) we find that This result can be interpreted as the propagation amplitude of one-particle states with an x-dependent normalization because The left handed-field ψ L can be further decomposed into its spin-up and spin-down components by means of the projection operatorŝ which separate the spin-up component and the spin-down component of the field along the third direction respectively. That is to say,   Similarly we use the symbol D ↑ L to denote the nonzero component ofp ↑ D Lp↑ . Following section 2 we know that the Feynman propagator S ↑(L) F (x, x ′ ) satisfies the following equations Eq.(5.17) implies that in the frame of reference (5.15) in which (x, x ′ ) corresponds to propagating modes confined to a two dimensional subspace. Since S + F (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 This can be also verified by using the path integral (1.1) and noting that in this frame of reference, Eq.(5.30) reads Following the argument in section 3.1 S ↑(L) F (x, x ′ ) in Eq.(5.42) can be interpreted as the propagation amplitude of one-particle states localized on a two dimensional Minkowski spacetime.
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. 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.

One particle states
Let ψ L be the left-handed component of a Dirac field. Suppose that we project ψ L into its "spin-up" component wherep ↑ is a projection operator whose Lorentz transformation is given by 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) and equations, The spin-up component of a left-handed fermion along the x 3 -direction is separated byp In flat spacetime, using Eq.(6.10) and Eq.(5.11) in Eq.(6.3) we obtain Thus the plane-wave solution of the classical field equation (6.6) is given by (6.12) up to a normalization constant, where x ± := x 0 ± x 3 , (6.13) x ⊥ := (x 1 , x 2 ), (6.14) and q ⊥ ∈ R 2 . Following Eqs.(6.8) and (6.9) 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 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.

The Kerr solution
The x 0 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 16 The argument is similar to footnote 15.
As an example, consider the Kerr solution whose line element in the Kerr coordinates is given by where ρ 2 := r 2 + a 2 cos θ 2 , ∆ := r 2 + a 2 − 2mr, and m and ma are constants representing the mass and the angular momentum as measured from infinity [21]. Since g rr = 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 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 g u − 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) are given by du ± = dt ± (r 2 + a 2 )∆ −1 dr, (6.34) dφ + = dφ + a∆ −1 dr. (6.35) and consequently u − = t − r + 2m ln r + O(r −1 ), (6.36) one verifies that the u − ordering of the Feynman propagator reproduces the ordinary t ordering via Eq.(6.30) asymptotically.

Conclusion
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 ψ ↑ L which is annihilated by a null vector field ∂ − in a certain frame of reference. In a coordinate system given by x ± := x 0 ± x 3 , and x ⊥ := (x 1 , x 2 ) such that the metric component g −− = 0, this frame of reference is identified by the following conditions on the tetrad e µ (a) , in which, σ 3 is the third Pauli matrix and ψ L is a left-handed massless Dirac field. The corresponding Feynman propagator is given by (7.4) in which S + F (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) F 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 spacetime. In the flat spacetime limit, ψ ↑ L is reminiscent of the spin-up component of a left-handed massless Dirac field travelling along the x 3 -axis.
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 S + F (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 where D L is the Dirac operator for massless fermions in the left-handed sector, and ψ ↑ L is the spin-up component of the left-handed Dirac field ψ L ψ ↑ L :=p ↑ ψ L , (7.6) in whichp ↑ is the corresponding projection operator. We have supposed that ψ ↑ L is in the spin 1 2 representation of the local Lorentz transformations similarly to ψ L . This can be done by consideringp ↑ as a tensor field. Explicitly, if U L (Λ) denotes the operator corresponding to a local Lorentz transformation Λ in the left-handed sector such that ψ L → U L (Λ)ψ L , we require that accordingly. In the frame of reference given by Eqs. Therefore, the classical field equation implies that ψ ↑ L is annihilated by the null vector ∂ − and the Feynman propagator is given by Eq. (7.4). Consequently, the notion of fermionic one-particle states and the corresponding vacuum state is well-defined in four dimensional (non-stationary) curved backgrounds. Such particles travel without being scattered by the background geometry.