Fermionic one-particle states in curved spacetimes
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
2 General properties of Feynman propagator
Massless Dirac field 3.1
Upside down approach 4.1 4.2 4.3
3 Two dimensional Minkowski spacetime
4 Two dimensional curved spacetime
5 Four dimensional curved spacetime
Conformally flat spacetimes
6 One particle states
6.1 The Kerr solution
– 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 .
In principle, the vacuum state can be inferred from a plausible two-point function of
the quantum fields . Such a two-point function can be computed by path integrals. For
Dirac spinors it is given by
SF (x, x′) = Z
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′
tively. That is, we suppose that3
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 ,
Lorentz transformation and the field equations (1.8) and (1.9) give
ψ(x) = X Z
ψ(x) = X Z
v¯s(p)e−ip·x hp~, s; B| ,
u¯s(p)eip·x |p~, s; Ai .
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 
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 –
The spinors anti-commute at spacelike separation, i.e.,
SA(x, x′) =
SB(x, x′) =
d3p p/ + m
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 .
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 .
∇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 .
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 . 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 ∂−,
) = 0,
– 3 –
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
S↑(L)(x, x′) = SF+(x, x′)δ2(x⊥ − x′ ),
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
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
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.
General properties of Feynman propagator
Our starting point is eq. (1.1) in which
S := i
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) = − √
S := −i
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 = − √
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 –
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
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
SF−(x, x′) !
SF±(x′, x) ⊜ −SF±(x, x′)∗.
SF(L)(x, x′) !
S(L)(x′, x) ⊜ −SF(L)(x, x′)†.
– 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
and PR := 1+γ5 are projection operator which
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
The corresponding field equation is
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,
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 –
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
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
u¯a(p)eip·x |A; p, ai ,
ua(p)e−ip·x hA; p, a| ,
SA(x, x′) :=
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 –
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
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)
one verifies that
Using the equality
ua(p)u¯a(p) = 2 |p| θ(−ap)σ−a,
SrBsa(x′, x) = SrAsa(x, x′) =
Z ∞ dp e−ip(xa−x′a)σr−sa.
dp e−ipx = πδ(x) − i
dp sin px,
in which, δ(x) := ddx θ(x) denotes the Dirac delta function, one obtains
δ(xa − x′ ) − 2π 0
i Z ∞
dp sin p(xa − x′a)
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,
Upside down approach
SFa (x, x′) =
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
2∂−aG(a)(x, x′) = δ2(x − x′),
a = ±
(a)(x, x′) ⊜ 1
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
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
|B; p, ai and subsequently postulate the vacuum state |0i.
Two dimensional curved spacetime
We denote the spacetime metric by gμν and the Minkowski metric by12
The local frame is identified by the tetrad eμ(a) and its inverse
gμν = eμ(+)eν (−) + eμ(−)eν (+),
where eμ(±) := √12 (eμ(0) ± eμ(
)). 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
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
As a result
where 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.
tively, eq. (4.9) reads
S = S+ + S−,
Sa := √
d2x Eμ(a)(x) ψa(x)∗ ←∂→μ ψa(x)
The action is given by  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
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 = ±
(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
which according to the Peano existence theorem has at least one solution only if Eμ(a)(x)
are continuous . 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
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,
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 = ±
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′ )
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
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
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
denotes the spin connection. Here,
D(e) := eμ(a)γa∂μ + Ω,
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
− 2 ξabΣab ,
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
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(+).
(ξ)eμ(a) := Λ(ξ)abeμ(b).
D((ξ)e) = U (ξ)D(e)U (ξ)−1.
℧μν (Ω) := ∇μΩν − ∇ν Ωμ − 2
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
In two dimensions,
(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
one verifies that D(e) = [[D+, D−]]/.
Ω = [[∇μeμ(−), ∇μeμ(+)]]/.
∂μEμ(a) = √
Eq. (4.45) implies that under a general coordinate transformation,
∂μEμa(x) → ∂μ′E′μ(a)(x′) = det
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
(+)(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
and consequently, eq. (1.1) implies that the Feynamnn propagator is given by
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+) . This theory
is the typical example of systems with gravitational anomaly [13, 18–20].
Four dimensional curved spacetime
Following  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 := −
In this notation, eq. (4.34) reads
Ωμ = [[ΩLμ, ΩRμ]]\,
in which ΩRμ := −
σ2Ω∗Lμσ2. Furthermore, eq. (4.36) reads
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,
2 ∇μeμ(a) +
Iμ := ǫabcdeμ(a)eρ(b)eν (c)∂ρeν (d),
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
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 ,
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) ,
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 .
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.
DL(x) = σ¯aδμae−ω(x) ∂μ +
2 ∂μω(x) .
Consider the two-point function
and xa := δμa xμ. Using the identity
we find that
Gc(fL)(x, x′) := i
p · σ
DL(x)Gc(fL)(x, x′) =
δ4(x − 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′)),
G1(Lp)(x, x′) := i
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
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
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,
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).
1 + σ
1 − σ
DL↑SF↑(L)(x, x′) = (−g(x))− 12 δ4(x − x′),
S↑(L)(x′, x) ⊜ −SF↑(L)(x, x′)∗.
DL↑ = 2e−ω(x) ∂− +
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
S↑(L)(x, x′) ⊜ SF+(x, x′)δ2(x⊥ − x′⊥)e− 32 (ω(x)+ω(x′)),
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
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
Now consider another frame of reference given by
eμ(−)dxμ = e4ωdx−,
eμ(a)dxμ = eωdxa,
a = 1, 2,
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′ ).
This can be also verified by using the path integral (1.1) and noting that in this frame of
reference, eq. (5.30) reads
S[ψL↑†, ψL↑] = 2i
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
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
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
One particle states
S[ψL↑†, ψL↑] := i
DL↑ψL↑ = 0.
S↑(L)(x, x′) = pˆ S↑(L)(x, x′)pˆ† ,
DL↑S↑F(L)(x, x′) =
δ4(x − x′)pˆ† ,
S↑(L)(x′, x) ⊜ −S↑F(L)(x, x′)†.
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 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
The spin-up component of a left-handed fermion along the x3-direction is separated
In flat spacetime, using eq. (6.10) and eq. (5.11) in eq. (6.3) we obtain
1 + σ
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
⊥ ∈ R2. Following eqs. (6.8) and (6.9) we obtain
S↑(L) = SF+(x, x′)δ2(x⊥ − x′⊥)pˆ↑,
gμν = (−g)− 12 eμ(+)g−ν + eν(+)g−μ −
a = −, 1, 2.
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,
These equations also show that for µ, ν
= +, 1, 2,
e−(+) = (−g) 2 ,
e−(a) = 0,
g−μ = (−g) 2 eμ(−),
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(−) =
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 ∂−.
S[ψL↑†, ψL↑] := 2i
d4xψL↑†∂ ψ↑ ,
cos ϕ − sin ϕ !
sin ϕ cos ϕ
∂−ϕ = e′ (
We also assume that
where ∇μ denotes the Levi-Civita connection. This requirement can be satisfied by using
local rotations in the ((
) − (
)) 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.
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 . 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 =
(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)
∂(−) = (−g)− 21 ∂−,
) = 0,
1 + σ3
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
du± = dt ± (r2 + a2)Δ−1dr,
dφ+ = dφ + aΔ−1dr ,
− = 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.
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′ ),
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 , 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,
S[ψL↑†, ψL↑] := 2i
S[ψL↑†, ψL↑] := i
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
accordingly. In the frame of reference given by eqs. (7.1) and (7.2), the projection operator
Therefore, the classical field equation implies that ψL↑ 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
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