Chiral kinetic theory in curved spacetime
Yu-Chen Liu Lan-Lan Gao Kazuya Mameda Xu-Guang Huang






Physics Department and Center for Particle Physics and Field Theory, Fudan University, 
 Shanghai 200433, China


Physics Department and Center for Particle Physics and Field Theory, Fudan University, 
 Shanghai 200433, China

mame@nt.phys.s.u-tokyo.ac.jp
Physics Department and Center for Particle Physics and Field Theory, Fudan University, 
 Shanghai 200433, China

huangxuguang@fudan.edu.cn
Physics Department and Center for Particle Physics and Field Theory, Fudan University, 
 Shanghai 200433, China
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Fudan University, 
 Shanghai 200433, China


Many-body systems with chiral fermions exhibit anomalous transport phenomena originated from quantum anomalies.
Based on quantum field theory, we derive the kinetic theory for chiral fermions interacting with an external electromagnetic field in a background curved geometry. The resultant framework is U(1) gauge invariant and local Lorentz and diffeomorphism covariant.
It is particularly useful to study the gravitational or non-inertial effects for chiral fermions. As the first application, we study the chiral dynamics in a rotating coordinate and clarify the roles of the Coriolis force and spin-vorticity coupling in generating the chiral vortical effect. We also show that the chiral vortical effect is an intrinsic phenomenon of a rotating chiral fluid, and thus independent of the observer's frame.


Chiral kinetic theory in curved spacetime
    Xu-Guang Huang
    December 30, 2023
=========================================







§ INTRODUCTION


Quantum anomaly is a prominent concept in the transport phenomena of chiral fermions. One of its most novel consequences is the generation of parity-breaking currents, typified by the chiral magnetic effect (CME) <cit.> and chiral vortical effect (CVE) <cit.>. A crucial feature of these anomalous currents is that they are insensitive to the details of interactions and are thus universal. For this reason, such phenomena have received a lot of attention in a wide context of physics ranging from the high-energy nuclear physics <cit.> and astrophysics <cit.> to condensed matter physics <cit.>.

To study the real-time dynamics of the anomalous transport phenomena, the chiral kinetic theory (CKT) is a promising approach which is applicable when the system is dilute and the external fields are weak <cit.>. In CKT, the chiral anomaly is encoded through the Berry curvature <cit.>, which modifies the Boltzmann equation and the phase space measure. Recently, various aspects of the CKT were investigated, including the Lorentz covariance <cit.>, consistent versus covariant anomalies <cit.>, particle collisions <cit.>, etc.

Despite these developments, so far the CKT is restricted to flat spacetime and thus not conventional to explore the anomalous transport phenomena induced by gravitational or non-inertial effects <cit.>. Although the classical Boltzmann equation is readily extended to curved spacetime, the formulation of the CKT in curved spacetime is highly nontrivial.
The very few attempts so far <cit.> considered only a special curved spacetime, that is, the rotating coordinate
 [In this paper, the rotating coordinate will be regarded as a curved spacetime even though its Riemann curvature is zero, while the term “flat spacetime” is specifically referred to as the Minkowski spacetime.],
and lacked the diffeomorphism (i.e., the general coordinate transformation) covariance.
The more rigorous derivation should start from quantum field theory in curved spacetime.

In this paper, we derive the CKT in an arbitrary curved spacetime and external electromagnetic field, based on the Wigner function formalism that respects the U(1) gauge invariance, and the local Lorentz and diffeomorphism covariance <cit.>. We apply the resultant framework to a rotating coordinate and examine the frame dependence of the CVE, which is so far unclear. We show that, depending on the observer's frame, the Coriolis force and spin-vorticity coupling (and the side-jump effect) can be responsible to the generation of the CVE, but the total CVE current is always independent of the observer's frame.

Throughout this paper, we choose the unit c=k_B=e=1 (with e the electric charge), but keep ħ explicit;
(un)hatted Greek indices denote local flat (curved) spacetime coordinates;
η_ = diag(1,-1,-1,-1) is the Minkowski metric;
∇_μ denotes the covariant derivative with respect to the diffeomorphism and local Lorentz transformation, e.g., for scalars ∇_μ f=∂_μ f, for vectors ∇_μ V_ν = ∂_μ V_ν  - Γ^λ_μνV_λ, for spinors ∇_μψ=(∂_μ + Γ_μ)ψ, where the spin connection is Γ_μ= -i/4σ^α̂β̂ g_ρσe^ρ_(∂_μ e^σ_ + Γ^σ_μνe^ν_), with the spin matrix σ^ = i/2[γ^,γ^], vierbein e^μ_, and the Christoffel symbol Γ_μν^ρ = Γ_νμ^ρ;

the Levi-Civita symbol is ε^=√(-g(x))ε^ with ε^ = -ε_ = 1 and g= (g_μν);
the Dirac matrices satisfy {γ^μ,γ^ν} = 2g^μν and γ^5 is defined as γ^5= (-i/4!)ε_μνρσγ^μγ^νγ^ργ^σ.





§ PHASE SPACE AND HORIZONTAL LIFT


In curved spacetime the definition of the phase space is subtle because a global notion of momentum is usually not permitted.
This means that for each position x on the spacetime manifold , one can introduce a momentum space _x attached to .
The phase space is then the collection of (x,_x), which constitutes a fiber bundle.
One of the natural choices for _x is the tangent or cotangent space so that the usual momentum space is reproduced in Minkowski spacetime.
In this paper, we employ the latter.
That is, we define the momentum variable p_μ as a point in the cotangent space (that is, p_μ is a covariant vector on ), and the corresponding phase space is the cotangent bundle ^*.
Similarly, a point y^μ in the tangent space _x (that is, y^μ is a contravariant vector on ) is defined as the position variable canonically conjugate to p_μ.
The set of (x,_x) builds the tangent bundle .

The Wigner function W(x,p) for Dirac fermions that we will introduce in next section is required to transform covariantly under the U(1) gauge transformation, local Lorentz transformation, and the diffeomorphism.
The U(1) covariance is ensured when W(x,p) is suitably constructed with ∂_μ + iA_μ/ħ, instead of ∂_μ, see next section and Ref. <cit.>. Consequently, W(x,p) is transformed as W(x,p)→ S(x)W(x,p)S^-1(x), where S is the representation matrix of  the U(1) gauge transformation.
In a similar manner, the local Lorentz covariance can be kept by introducing the spin connection Γ_μ and replacing ∂_μ by the covariant derivative ∇_μ; W(x,p) is thus transformed as a bispinor, W(x,p)→ U(Λ)W(x,Λ p)U^-1(Λ), where Λ is the local Lorentz transformation at x and U(Λ) is its spinorial representation.
The diffeomorphism covariance of W(x,p) needs more careful treatment.
This is because the diffeomorphism affects functions in  or ^* in a very nontrivial way.
In fact, the proper covariant derivative on ^* is defined as follows <cit.>:

    D_μ = ∇_μ
     + Γ^λ_μνp_λ∂^ν_p

with ∂^μ_p=∂/∂ p_μ.
The derivation of Eq. (<ref>) based on the parallel displacement is shown in Appendix <ref>.
In differential geometry, such defined D_μ is called the horizontal lift of ∇_ on 𝕄 to ^*.
Similarly to Eq. (<ref>), the covariant derivative in  is defined by D_μ = ∇_μ - Γ^λ_μν y^ν∂^y_λ.


The implementation of the horizontal lift brings a great advantage in analysis. That is, we can regard p_ and y^ as “x-independent" variables under the parallel transport by D_, because of

    D_μ p_ν = D_μ y^ν = 0  .

As a result, for an arbitrary function Ψ(x) on 𝕄, its lifted image in  is represented as the function translated by D_μ:
Ψ(x,y) ≡Ψ(x)+y^μ∇_μΨ(x) +1/2y^μ y^ν∇_μ∇_νΨ(x)+⋯ = exp(y· D)Ψ(x).
Furthermore, the Fourier transformation from  to ^* is expressed as
Ψ(x,p)= ∫ d^4y √(-g(x)) exp(-ip· y/ħ) Ψ(x,y).









§ QUANTUM TRANSPORT IN CURVED SPACETIME


With the above preparation, we define the fermionic Wigner function covariantly under the U(1) gauge, local Lorentz transformations, and diffeomorphism, as follows:

    W(x,p)     = ∫ d^4y √(-g(x)) e^-ip· y/ħ ρ(x,y) , 
    ρ(x,y)     = ⟨ψ̅(x,y/2) ⊗ψ(x,-y/2)⟩

with ψ(x) being the Dirac spinor on , ψ̅(x)≡ψ^†(x)γ^, ψ̅O≡ [O ψ]^†γ^ for an operator O, ψ(x,y) =  exp(y· D)ψ(x), ψ̅(x,y) = ψ̅(x)exp(y·D), and [ψ̅⊗ψ]_ab=ψ̅_b ψ_a (a,b=1-4).
Note that D_μ acting on the Dirac spinor involves A_μ to keep the U(1) gauge covariance:

    D_μψ(x,y)
     = (
     		∇_μ
     		- Γ^λ_μν y^ν∂^y_λ
     		+ iA_μ/ħ) ψ(x,y)  ,

where we recall that ∇_μψ further involves the spin connection, i.e., ∇_μψ=(∂_μ + Γ_μ)ψ. 
In Minkowski spacetime, Eq. (<ref>) is reduced to a simple form with the Wilson line <cit.>:
ρ(x,y)=⟨ψ̅(x_+)⊗ Pexp[-iħ∫_x_-^x_+dz· A(z)] ψ(x_-)⟩ with x_±=x± y/2 and P the path ordering symbol.
In this paper, we focus on the collisionless fermions, so the spinor field obeys the Dirac equation

    γ^μ(∇_μ + iA_μ/ħ) ψ(x)
     = ψ̅(x) (∇_μ - iA_μ/ħ)γ^μ
     = 0 .




Computing D_μρ(x,y) and ∂_μ^yρ(x,y) with the help of the Dirac equation (<ref>), we derive:

    γ^μ(p_μ + iħ/2D_μ)  W
      = iħγ^μ∫ d^4y √(-g(x)) e^-ip· y/ħ
           ×⟨ψ̅(x,y/2) ⊗ (-_μ + _μ)ψ(x,-y/2) 
                        -ψ̅(x,y/2)_μ⊗ψ(x,-y/2)
       ⟩ ,
          _μψ(x,y)
      = -iy^ν/ħ∑_n=0^∞[(y· D)]^n/(n+1)! G_μν ψ(x,y)  ,
          _μψ(x,y)
      = -iy^ν/2ħ∑_n=0^∞[(y· D)]^n/(n+2)!  G_μν ψ(x,y) ,

where (X)Y ≡[X,Y] represents a commutator. The details of the derivation of Eq. (<ref>) can be found in Appendix <ref>; see also Ref. <cit.>.
We defined G_μν≡ -iħ[D_μ,D_ν] as the total curvature tensor on  or ^*.
For instance, we have G_μν ψ(x,y) = (F_μν +ħ/4R_μναβσ^αβ -iħR^ρ_σμνy^σ∂^y_ρ) ψ(x,y), where the Riemann tensor is R^ρ_σμν = 2∂_[νΓ_μ]σ^ρ +2Γ_λ[ν^ρΓ^λ_μ]σ with X_[μY_ν] = 1/2(X_μ Y_ν - Y_ν X_μ).
The transport equation (<ref>) involves the full quantum correction coupled with electromagnetic field and curved background.
In Minkowski spacetime, Eq. (<ref>) reproduces the transport equation derived in Ref. <cit.>.

In practice, Eq. (<ref>) is a powerful tool for the semiclassical analysis with the systematic expansion in terms of ħ.
Let us adopt the power counting scheme with p_μ = O(1) and y^μ∼ iħ∂^μ_p = O(ħ).
After a lengthy but straightforward calculation (see Appendix <ref>), the transport equation for the Wigner function up to O(ħ^2) is written down, as follows:
 [We actually keep the O(ħ^3) terms, which are necessary to derive Eqs. (<ref>)-(<ref>) at O(ħ^2).]

    γ^μ(Π_μ + iħ/2Δ_μ)W  
        = iħ^2/32γ^μ (
       			 R_μναβ
      		+ iħ/6∂_p·∇ R_μναβ)
        		  ∂_p^ν[ W, σ^αβ]  ,

with

    Π_μ   = p_μ
       	- ħ^2/12(∇_ρ F_μν)∂^ν_p∂^ρ_p
      	+ ħ^2/24R^ρ_σμν∂^σ_p∂_p^ν p_ρ
      	  +ħ^2/4R_μν∂_p^ν , 
    Δ_μ    = ∇_μ +(-F_μλ
      	 + Γ_μλ^ν p_ν)∂_p^λ
      	 - ħ^2/12(∇_ρ R_μν)∂_p^ρ∂_p^ν
           - ħ^2/24(∇_λR^ρ_σμν)
      	    ∂_p^ν∂_p^σ∂_p^λ p_ρ
      	 + ħ^2/8R^ρ_σμν∂_p^ν∂_p^σ D_ρ
          + ħ^2/24 ( ∇_α∇_β F_μν
      	  + 2R^ρ_αμνF_βρ )
      	    ∂_p^ν∂_p^α∂_p^β ,

where R_μν=R^ρ_μρν is the Ricci tensor.
Further we decompose Eq. (<ref>) with the basis of the Clifford algebra:
W =14[ +γ^5+γ^μ_μ+γ^5γ^μ_μ+12σ^μν_μν].
Then we obtain

    Δ·
     = ħ^2/24(∇_ρ R_μν) ∂_p^ρ∂_p^μ^ν ,
       Π·
     =ħ^2/8R_μν∂_p^μ^ν ,
       ħΔ_[μ_ν]
    -ε_μνρσΠ^ρ^σ
    =-ħ^2/16ε_μναβR^αβρσ∂_ρ^p_σ ,

with _μ = (_μ+_μ)/2 (see Appendix <ref> for the derivation).
The first equation will be the kinetic equation for right-handed Weyl fermions, while the second and third serve as constraints.
The equations for _μ=(_μ-_μ)/2 are the same, except for a sign change in front of the first term of Eq. (<ref>).










§ CHIRAL KINETIC EQUATION AT O(Ħ)


Now we focus on the kinetic equation for ^μ(x,p) at O(ħ).
Equations (<ref>)-(<ref>) are reduced to

    Δ· = 0  , 
        p· = 0 , 
       ħΔ_[μ_ν]
      - ε_μνρσ p^ρ^σ = 0

with Δ_μ = ∇_μ +(-F_μλ + Γ_μλ^ν p_ν)∂_p^λ.
Thanks to the horizontal-lift prescription, we can solve Eqs. (<ref>)-(<ref>) in the same manner as that in flat spacetime.
The general solution is given by <cit.>

    ^μ
     = 4πδ(p^2)[
     p^μ -ħ/p^2^μνp_ν
     + ħΣ_n^μνΔ_ν]f + O(ħ^2)

with ^μν = ε^μνρσF_ρσ/2 and f=f(x,p) being the distribution function.
The last term is called the side-jump term <cit.>;
we introduced the spin tensor Σ_n^μν≡ε^μνλρ p_λ n_ρ/ (2 p· n), where n^μ(x) is an arbitrary vector to satisfy n· p≠ 0 and n^2=1.
This vector field accounts for an ambiguity in defining the spin for massless particles <cit.>.
Different n^μ's correspond to different spin-frames and they are connected via n'^μ = L^μ_ν n^ν = e^  μ_Λ^_ e^_  ν n^ν with Λ^_ being a matrix representation of the local Lorentz transformation.

Plugging Eq. (<ref>) into Eq. (<ref>), we eventually obtain

    δ(p^2-ħ F_αβΣ_n^αβ)[
     p·Δ+ħ(n_μ^μν/p· n
      + Δ_μΣ^μν_n
     )Δ_ν
               
     + ħ/2Σ^μν_n (∇_ρ F_μν
     - p_λR^λ_ρμν) ∂_p^ρ] f = 0  .

This is the curved-spacetime generalization of the conventional chiral kinetic equation <cit.>.
Several comments are in order.
(I) In the classical limit ħ→ 0 we reproduce the Einstein-Vlasov equation:
δ(p^2)  p^μ[∂_μ +(- F_μν + Γ_μν^λ p_λ)∂^ν_p] f = 0.
(II) The spin connection Γ_μ is unrelated to Σ_n^μν.
Indeed, since ^μ is a vector, such a connection can never appear in Eq. (<ref>).
(III) The Riemann curvature naively seems to be an O(ħ^2) correction, as Eqs. (<ref>)-(<ref>) show.
However, once coupled with the side-jump term, it emerges even at O(ħ) in Eq. (<ref>).
This term represents the so-called spin-curvature force <cit.> for chiral fermions.
(IV) On the other hand, the curvature does not appear in the delta function, which designates the on-shell condition.
However, this would not be the case at O(ħ^2).
In fact, from the viewpoint of field theory, the dispersion relation (without U(1) gauge field) reads p^2 - ħ^2R/4 = 0 due to (-iħγ^μ∇_μ)^2ψ = -ħ^2(∇_μ∇^μ+R/4)ψ = 0 <cit.>
(see also Ref. <cit.> for a curvature correction to the CVE).





§ EQUILIBRIUM STATE


To reveal the physical content of Eq. (<ref>), we consider the equilibrium state. We drop A_μ for simplicity.
At equilibrium, f is generally written as a function of the linear combination of the collisional conserved quantities, i.e., the particle number, the linear momentum, and the angular momentum.
Therefore we have f= (g) 
with g=α(x)+β^μ(x)p_ +ħγ_μν(x)Σ^μν_n.
Note that the orbital angular momentum is involved in the second term.
Plugging (g) into Eq. (<ref>) and requiring it to hold for arbitrary p_μ, we arrive at the following constraints:


    ∇_μβ_ν + ∇_νβ_μ = g_μνϕ(x)  ,
       ∇_μα = 0  ,  γ_μν = 1/2∇^⊥_[μβ_ν] ,

where ϕ is an arbitrary scalar function and ⊥ represents the component perpendicular to n^μ.
In Appendix <ref> we present the derivation of Eq. (<ref>) and Eq. (<ref>).

We have three comments about the above equations.
(I) Equation (<ref>) is the conformal Killing equation.
Choosing a timelike β^μ, we define the fluid velocity and temperature via β^=U̱^μ (with U^2=1) and T=1/$̱, respectively. The physical meaning ofϕis the expansion rate of the fluid:ϕ= 1/2∇·β, which follows from Eq. (<ref>).
Thus the fluid is kept equilibrium under such an expansion.
This is understood as the conformal invariance in the massless Dirac theory.
Note that for massive particlesϕmust vanish, as the expansion can drive the system out of equilibrium.
(II) From Eq. (<ref>), we find thatαis a constant scalar.
We define the chemical potential throughα= -βμ.
(III) The equilibrium distribution is eventually given byf=(g)with

    g = β(-μ + p· U)
       + ħ/2Σ^μν_n∇_μ(β U_ν) .

The last term expresses the spin-vorticity coupling.




§ ROTATING COORDINATE

As the first application, we use our framework to revisit the derivation of the CVE by considering a rotating coordinate.

Let us choose a constant angular velocity= (ω^1,ω^2,ω^3)and hereafter setA_μ=0.
The corresponding metric tensor reads

    g_00 = 1 - ^2 ,    g_0i = u^i ,   g_ij = -δ_ij










with= (u^1,u^2,u^3) = ×.
The nonzero components of the Christoffel symbol areΓ_00^i = -x^i^2 +(·)ω^i = (×)^iandΓ_0j^i = Γ_j0^i = -ε^ijkω^k(withε^123=1), which lead toR^ρ_σμν = 0.
The metric has an infinite red-shift surface at distancer=1/||away from the rotating axis.
We focus on the spacetime region inside this surface, and thus ignore the boundary effect of the system.
Such an assumption works as long as the angular velocity is small enough compared with other characteristic scales of the system <cit.>.
Therefore in the following analysis of the CVE, we consider the slowly rotating coordinate with|| ≪Tor|| ≪μ.

In this case, the metric (<ref>) admits two timelike Killing vectors;K^μ_in=g^μ0andK_rot^=^̣_0.
Note that the former (latter) corresponds to the inertial (rotating) observers
 [To be more specific, the rotating (Minkowski) coordinate is considered as the coordinate chart of the rotating (inertial) observer.].
The velocities of these two observers are

    ^μ =(1,)  ,  ^μ =(g_00)^-1/2δ_0^μ ,

which are normalized as𝒰^2 = 1.
From the on-shell conditionp^2 = g^μνp_μp_ν= 0, we obtain

    ϵ_≡ K_ rot^μ p_μ=p_0
     = ||+· , 
       _ = ∂ϵ_/∂ =  +  ,

where_denotes the group velocity and the three-momentum is defined as= -(p_1, p_2, p_3).

Similarly, we can obtainϵ^in_≡K_in^μp_μ=p^0 = ||and^in_=∂ϵ^in_/∂= . Thusis identified as the momentum observed by the inertial observer. Note that the second terms in Eqs. (<ref>) and  (<ref>) correspond to the rotating energy and velocity shifts, respectively.
In the following, we analyze the CKT with several choices ofn^μandU^μ.





 §.§ Inertial fluid

First of all, we consider an inertial fluid (i.e., a fluid at rest in flat spacetime) with a rotating observer.
We setU^μ= n^μ=^μandK^μ=K^μ_rot.
Performing thep_0-integration of Eq. (<ref>), we find (for the particle channel only; antiparticle channel is similar)

    [
      ∂/∂ t + _·∂/∂
      +(×)·∂/∂] f(t,,) = 0

withf(t,,) = f(t,,,p_0=ϵ_).
From the above equation, we identify = _and = ×, which reproduce the Coriolis and centrifugal force: = 2×-×(×).
From Eq. (<ref>), the particle number current reads







    J^μ = (J^0 ,)
        ≡∫d^4 p/(2π)^4√(-g(x))^μ ,
        = [_
       -ħ||_×∂/∂] f(t,,)

with= ∫d^3p (2π)^-3and_= /(2||^2)being the Berry curvature.
Note that due to∇_μ^ν= 0, at equilibrium all theO(ħ)corrections disappear in Eq. (<ref>), and thus it is just the classical Liouville current:= _.

Also from Eqs. (<ref>) and (<ref>) forU^μ= ^μ, we can check that the same is true for arbitraryn^μ.

Therefore, the CVE is never induced by an inertial fluid, independently of the observer's reference frame and the spin-frame choosing vectorn^μ.





 §.§ Rotating fluid

In this case we adoptU^μ= ^μ, that is, we consider a fluid at rest in the rotating coordinate.
Hereafter let us focus on the smallωlimit to simplify the discussions.

First, we choosen^μ= ^μ, which leads to the kinetic equation and the current as the same forms as Eq. (<ref>) and (<ref>), respectively.
However, physical quantities are affected by quantum corrections.







When we takef=(g) = 1/(e^g+1)and append the antiparticle contribution (for whichμis replaced with-μ), theO()terms in Eq. (<ref>) yield

    _CVE
     = ħ(μ^2/4π^2 + T^2/12) ,

which is the well-known CVE current. The spin-vorticity coupling term in Eq. (<ref>) are prominent to induce_CVE.
Because of this coupling, the first term in Eq. (<ref>) gives1/3of_CVE, while the second yields2/3<cit.>.

Second, we employn^μ= ^μ.
It is more convenient to work with a new three-momentum defined as= (p^1,p^2,p^3)[
This is the change of the phase space variables that yields a nontrivial Jacobian which calls for ∂_μ→∂_μ + (∂_μ p^ν)∂_ν^p.
]
, whose physical meaning will be explained later.
After thep_0integration, the kinetic equation reads

    [
      (1+2ħ ||·_)∂/∂ t
      + {_
      + 2ħ||(_·_) }·∂/∂
           
      + 2||(_×)·∂/∂] f(t,,) = 0

with the modified velocity_= ∂ϵ̃_/∂and energy dispersion

    ϵ̃_ = || - ħ/2· .


The above kinetic equation exhibits an analogy between magnetism and rotation under two types of the correspondence, i.e.,||↔in_(and so in_), and2||↔elsewhere, reflecting the fact that the Landégfactor is2for spin-1/2 particles.
In other words, the spin-vorticity coupling plays a role of the magnetization coupling, and the Coriolis force can be regarded as a fictitious Lorentz force. This suggests thatis the momentum observed by the rotating observer.
Indeed, Eq. (<ref>) shows that the classical dispersion is linear to||. For this reason, Eq. (<ref>) are represented only with quantities in the rotating coordinate.
We note that the factor in front of∂/∂tin Eq. (<ref>) represents the quantum modification to the phase space measure <cit.>.

From Eq. (<ref>), we compute the particle number current as

    = ∫_[
      _ + 2ħ||(_·_) ] f(t,,) + O(ω^2) ,

which, once substituted withf=(g) = 1/(e^g+1), reproduces Eq. (<ref>) again.
Note that the first term does not contribute to the CVE current.
In other words, the Coriolis force is responsible for generating the CVE whereas the spin-vorticity coupling is not.
This explains why the heuristic replacement→2||works correctly <cit.> in computing the CVE current.

Some comments are in order.
(I) The above analysis shows that the origin of the CVE can be interpreted differently for differentn^μ.
For the inertial (rotating) spin-frame vectorn^μ=^μ(^μ), the CVE is induced through the spin-vorticity coupling (the Coriolis force). This is a clear demonstration for the nature of spinning massless particles: the total angular momentum is frame-dependently decomposed into the spin and the orbital parts <cit.>.


(II) However, in both cases withn^μ= ^μandn^μ= ^μ, we derive the same CVE current (<ref>).
Indeed, the choice ofn^is superficially irrelevant to the CVE, as it is compensated by the side-jump effect.
This is confirmed from the fact that for arbitraryn^μ, the equilibrium current is derived as a spin-frame-independent form:
forf=(g)=^+ ħ/2Σ_n^μν∇_μβ_ν(d^/dg)+O(ħ^2)with^= f(g=-βμ+ p·β), Eq. (<ref>) is reduced to

    J^μ_eq
    
    
    
    
    
    
    
           = ∫d^4p   2 δ(p^2)/(2π)^3√(-g(x))[ p^μ
       -ħβω^μ/2 (p· U) d/dg]^ ,

withω^μ=1/2^μνρσU_ν∇_ρU_σ.

At the same time, we note that Eq. (<ref>) also holds for arbitrary curved spacetime.
This explains why the CVE current (<ref>) is the same as that in Minkowski coordinate <cit.>.
The CVE is hence intrinsic for rotating fluid, of which the velocity configuration satisfiesω^μ≠0.







§ SUMMARY AND OUTLOOK

We extended the framework of the chiral kinetic theory (CKT) to curved spacetime, based on quantum field theory.
The CKT in curved spacetime is a primary tool for non-equilibrium chiral dynamics under the general-relativistic effect.

This enables us to investigate the anomalous transport phenomena in various chiral matter systems with (effective) gravitational field or non-inertial forces, such as supernova or neutron star environment <cit.>, rotating/expanding quark-gluon plasma <cit.>, thermal systems with the temperature gradient <cit.>, and Weyl/Dirac semimetals under strain <cit.> or possibly torsion <cit.>.

As an application, we analyzed the CKT in a rotating coordinate, and clarified the frame-dependent interpretation for the chiral vortical effect (CVE).
Our calculation showed that although the CVE receives contributions from both the spin-vorticity coupling and Coriolis force depending on the choice of the defining frame of spin, their sum is independent of both the observer's frame and the spin-frame.
In this paper, we did not discuss about the relation between the finite-temperature term in the CVE current and the gravitational anomaly <cit.>.

On the other hand, it is still left open if such a term is induced by the global anomaly <cit.>.
The CKT in curved spacetime is an auspicious candidate to lead to a model-independent answer to this mystery.
This will be shown in a future publication.



We thank Omer Faruk Dayi, Tomoya Hayata, Yoshimasa Hidaka, Bei-Lok Hu, Kristan Jensen, Jinfeng Liao, Qun Wang, and Yi Yin, for useful discussions and valuable comments.
This work is supported by the China Postdoctoral Science Foundation under grant No. 2017M621345 (K. M.), and the Young 1000 Talents Program of China, NSFC through Grants Nos. 11535012 and 11675041 (X.-G. H.).


§ HORIZONTAL LIFT

We derive the proper covariant derivative for functions in the cotangent bundle^*from the viewpoint of the parallel displacement <cit.>. First, suppose thatΦ(x,p)is a scalar function on^*.
Under the infinitesimal diffeomorphismx^μ→x'^μ= x^μ+ dx^μ, the variationδΦinvolves two parts: one comes from thex-dependence ofΦand the other from thep-dependence.
That is, we write

    δΦ
       
           = δ_xΦ + ∂^μ_p Φδ p_μ ,

whereδp_μis the variation ofp_μunder this diffeomorphic transformation.
By definition, we haveδΦ= 0andδp_μ= Γ_μν^ρp_ρdx^ν, which leads to

    δ_xΦ = -Γ_μν^ρ p_ρ dx^ν .

Now we define a derivativeD_μas

    D_μΦ dx^μ≡Φ(x+dx,p)-[Φ(x,p) + δ_xΦ(x,p)]  .

Thus we obtain

    D_μΦ = (∂_μ + Γ_μν^ρ p_ρ∂_p^ν) Φ ,

which is the horizontal lift of∂_μto the cotangent bundle for a scalar field.
In the same manner, the covariant derivative for an arbitrary tensor field in the^*is derived as

    D_μ = ∇_μ + Γ_μν^ρ p_ρ∂_p^ν ,

with∇_μthe usual covariant derivative on. Similarly, we can define the horizontal lift of∇_μto the tangent bundleas given in the main text.







§ IDENTITIES OF THE DIRAC MATRICES

We present some identities of the Dirac matrices which are useful in the derivation of the chiral kinetic equations.
From the definition { γ^μ,γ^ν} = 2g^μνandγ^5 = (-i/4!)ε_μνρσ γ^μγ^νγ^ργ^σ, we find



    γ^5 σ^μν = i/2ε^μνρσσ_ρσ ,
       γ^μγ^νγ^ρ
     = g^μνγ^ρ + g^νργ^μ  - g^ρμγ^ν
     - iε^μνρσγ^5γ_σ,

whereσ^μν = (i/2)[γ^μ,γ^ν].
From these relations, we can prove the following useful identities:

    γ^μγ^ν
      = g^μν -iσ^μν ,
       γ^μ[γ^ν, σ^αβ]
      = -4i g^μ[αg^β]ν
      	- 4 σ^μ[αg^β]ν , 
       γ^μγ^5 γ^ν
      = -g^μνγ^5
      	- 1/2ε^μναβσ_αβ , 
       γ^μ[γ^5γ^ν, σ^αβ]
      = 4iγ^5 g^μ[α g^β]ν
      	+ 2iε^ρσμ[αg^β]νσ_ρσ.


§ TRANSPORT EQUATION FOR W(X,P)

The Wigner function of Dirac fermions is defined by

    W(x,p) = ∫_y ρ(x,y) ,    ρ(x,y) = ⟨ψ̅_+ ⊗ψ_-⟩ .

Here we have introduced the following notations:∫_y = ∫d^4y √(-g(x)) e^-ip·y/ħ,ψ_- = ψ(x,-y/2) = e^-y·D/2ψ(x)andψ̅_+ = ψ̅(x,y/2) = ψ̅(x)e^y·/2.
Note thatD_μacting on the Dirac spinor involvesA_μto keep theU(1)gauge covariance, that is,

    D_μψ(x,y)
     = (
     	∇_μ - Γ^λ_μν y^ν∂^y_λ + iA_μ/ħ)ψ(x,y)  ,

with∇_μψ(x)=(∂_μ+ Γ_μ)ψ(x)andΓ_μthe spin connection. We consider the free Dirac field operators that obey

    γ^μ(∇_μ + iA_μ/ħ) ψ(x)
     = ψ̅(x) (∇_μ - iA_μ/ħ)γ^μ
     = 0 .

We assume that the surface integral for the Wigner transformation vanishes, that is,

    0
           = ∫ d^4 y ∂_μ^y [√(-g(x))e^-ip· y/ħρ(x,y) ] 
        =  -ip_μ/ħ W
      		+ ∫_y ∂_μ^y ρ(x,y)  .

In order to calculate the second term, we utilize

    D_μψ(x,y) = e^y· DD_μψ(x) - _μψ(x,y)  , 
       ∂^y_μψ(x,y) = D_μψ(x,y) + 2_μψ(x,y)  ,

which are derived from the operator identitye^Y X e^-Y = e^𝒞(Y)X with𝒞(Y) Z ≡[Y,Z].
Here_μare_μare defined as

    _μψ(x,y)
      = -iy^ν/ħ∑_n=0^∞[(y· D)]^n/(n+1)! G_μν ψ(x,y)  , 
       _μψ(x,y)
      = -iy^ν/2ħ∑_n=0^∞[(y· D)]^n/(n+2)!  G_μν ψ(x,y) .

HereG_μν ≡-iħ[D_μ,D_ν]is the total curvature tensor onand^*, e.g.,

    G_μνψ(x,y)
      = [ H_μν
      -iħR^ρ_σμνy^σ∂^y_ρ] ψ(x,y)  , 
        H_μν
      	=  F_μν +ħ/4R_μναβσ^αβ ,   
      F_μν
      	= 2∂_[μ A_ν] ,   R^ρ_σμν
      	= 2(
      		∂_[νΓ^ρ_μ]σ
      		+ Γ_λ[ν^ρΓ_μ]σ^λ)

with X^[μY^ν] = 1/2(X^μ Y^ν - X^ν Y^μ).
Then we compute

    ∫_y ∂_μ^y ρ(x,y)
       = 1/2D_μ W
      	  + ∫_y
      	  		⟨ψ̅_+ ⊗ (_μ -_μ)ψ_-
      	  			+ ψ̅_+ _μ⊗ψ_-
      	 		⟩
     	  - ∫_y
      	  		⟨ψ̅_+ ⊗ e^-y· D/2 D_μψ(x)
         		⟩ .

Contracted with γ^μ, the last term vanishes due to the Dirac equation (<ref>), and thus we derive

    γ^μ(p_μ + iħ/2D_μ)  W
        = iħγ^μ∫_y ⟨ψ̅_+ ⊗ (-_μ + _μ)ψ_-
       -ψ̅_+ _μ⊗ψ_-
       ⟩ .

This is Eq. (<ref>) in the main text.


§ SEMICLASSICAL EXPANSION UP TO O(Ħ^3)

Now we perform the semiclassical expansion of the transport equation (<ref>) (that is, Eq. (<ref>) in the main text) with the power counting scheme

    p_μ = O(1) ,   y^μ∼ iħ∂^μ_p = O(ħ).

It should be noticed thatD_μcan lead to terms ofO(ħ^-1)only when acting onψ(x,y):

    D_μψ(x,y) = O(ħ^-1) .

Since(y·D)ψ(x,y)is the same type function asψ(x,y)(i.e., a diffeomorphism scalar, local-Lorentz spinor), we readily find

    [(y· D)]^n G_μνψ(x,y)
      = [(y· D)]^n [ H_μν
      -iħR^ρ_σμνy^σ∂^y_ρ] ψ(x,y)  .

Then [(y· D)]^n G_μνψ(x,y) is calculated as follows:

    (y· D)G_μνψ(x,y)
         =   [
     		y· D
     		(
     			H_μν
     			-iħR^ρ_σμνy^σ∂^y_ρ)
     		+ i ħR^ρ_σμν y^σ D_ρ]  ψ(x,y)   , 
     [(y· D)]^2 G_μνψ(x,y)
        =   [
       (y· D)^2 F_μν
       + 2 iħ (y· D)R^ρ_σμν y^σ D_ρ
       - y^α y^σR^ρ_σμνF_αρ] ψ(x,y) + O(ħ^3)  .

Note that [(y· D)]^n G_μνψ(x,y) for n≥ 3 does not yield any O(ħ^2) contribution.
































Thus we obtain

    iħ∫_y
      	⟨ψ̅_+ ⊗ (-_μ + _μ)ψ_-
      		-ψ̅_+ _μ⊗ψ_-
      	⟩
       =   iħ/8[
      			3 H_μν∂_p^ν W
      			+ W _p^ν H_μν]
      	  + ħ^2/48[
      	  		5 ∂_p·∇ H_μν∂_p^ν W
      			- W _p^ν_p·∇ H_μν] 
          
      	  - iħ^3/48[
      			∇_α∇_β F_μν∂_p^α∂_p^β
       			- R^ρ_σμνF_αρ∂_p^α∂_p^σ] ∂_p^ν W
      	+ ħ^2/8R^ρ_σμν∂_p^ν
      			( 3 ^σ_ρ - ^σ_ρ )
      	 - iħ^3/48∂_p·∇R^ρ_σμν∂_p^ν
       			( 5 ^σ_ρ + ^σ_ρ ) 
          
      	+ iħ^3/48R^ρ_σμν∂_p^ν∂_p^σ
      		 ( 5 _ρ + _ρ )
      	 + ħ^4/192∂_p·∇R^ρ_σμν∂_p^ν∂_p^σ
      		 	( 7 _ρ - _ρ)
      	 + O(ħ^4)  ,

In the above equation we have defined

    ^μ_ν
      = ∫_y y^μ⟨ψ̅_+⊗∂^y_νψ_- ⟩ ,   ^μ_ν
      = ∫_y y^μ⟨ψ̅_+ ^y_ν⊗ψ_-⟩ , 
       _μ
      = ∫_y ⟨ψ̅_+⊗ D_μψ_- ⟩ ,   _μ
      = ∫_y ⟨ψ̅_+ _μ⊗ψ_-⟩ .

Although each of these is not a simple expression, their combinations are reduced to

    ^σ_ρ - ^σ_ρ
       = -iħ/2∂_p^σ D_ρ W
      		- iħ/4 F_λρ∂_p^λ∂_p^σ W
      		+ O(ħ^2)  ,    ^σ_ρ + ^σ_ρ
       = -∂^σ_p p_ρ W  , 
       _ρ - _ρ
       = -2ip_ρ/ħ W + O(ħ)  ,     _ρ + _ρ
       = D_ρ W ,

which follow from Eq. (<ref>).
From these relations and Eq. (<ref>), we eventually derive

    γ^μ(Π_μ + iħ/2Δ_μ) W
       =  i/4γ^μΘ_μαβ[ W, σ^αβ]

with

    Π_μ   =    p_μ
       	- ħ^2/12(∇_ρ F_μν)∂^_p∂^ρ_p
      	+ ħ^2/24R^ρ_σμν∂^σ_p∂_p^ν p_ρ
      	  +ħ^2/4R_μν∂_p^ν , 
    Δ_μ   =   ∇_μ +(-F_μλ
      	 + Γ_μλ^ν p_ν)∂_p^λ
      	 - ħ^2/12(∇_R̊_)∂_p^∂̊_p^ν
      	 - ħ^2/24(∇_łR^ρ_σμν)
      	    ∂_p^ν∂_p^∂_p^λ p_ρ
          
      	 + ħ^2/8R^ρ_σμν∂_p^ν∂_p^σ D_ρ
      	 + ħ^2/24 ( ∇_α∇_β F_μν
      	 + 2R^ρ_αμνF_βρ )
      	    ∂_p^ν∂_p^α∂_p^β ,  
    Θ_μαβ   =   ħ^2/8 R_μναβ∂_p^ν
      		+ iħ^3/48∂_p·∇ R_μναβ∂_p^ν .

They are Eqs. (<ref>) and  (<ref>) in the main text.


§ SPINOR DECOMPOSITION

In the last step, we decompose Eq. (<ref>) with the basis of the Clifford algebra.
The Wigner functionWis written as

    W = 1/4[
     			 + γ^5
     			+ γ· + γ^5γ·
     			+ 1/2σ^μν_μν]  .

The coefficients_μand_μcorrespond to the vector and axial currents, respectively:tr[γ^μW] = ^μandtr[γ^μγ^5 W] = ^μ.
This means that the right- and left-handed currents are represented as

    _μ = 1/2(_μ+_μ) ,  _μ = 1/2(_μ-_μ) .

Due to the masslessness in Eq. (<ref>), the vector and axial channels are decoupled from the scalar, pseudo-scalar, and tensor ones. Focusing on the vector and axial channels, we find

    γ·Λ (γ· + γ^5 γ·)
         = i/4γ^μΘ_μαβ[
        			γ· + γ^5 γ·,
        			σ^αβ] ,

where we have introduced the shorthand notationΛ_μ= Π_μ+ (iħ/2) Δ_μ.
By inserting Eqs. (<ref>)-(<ref>) into Eq. (<ref>) and extracting three parts proportional to1,γ^5andσ^μν, respectively, we obtain

    (Λ_ν - Θ^μ_μν) ^ν = 0  , 
        (Λ_ν - Θ^μ_μν) ^ν = 0  , 
       Λ_[μ_ν] - Θ_[μν]ρ^ρ
        - i/2ε_μνρσ (
        Λ^ρ^σ
        - Θ^ρσλ_λ  )
        = 0  .

Note that, by contracting(i/2)ε^μν_    αβwith the third equation, we obtain the same equation with the replacement^μ↔^μ.
Separating the real and imaginary parts, we finally derive

    Δ·
      = ħ^2/24∂_p·∇ R_μν∂_p^μ^ν , 
       Π· = ħ^2/8R_μν∂_p^μ^ν , 
       ħΔ_[μ_ν]
     - ε_μνρσΠ^ρ^σ
     = -ħ^2/16ε_μνρσ
      R^αβρσ∂_α^p_β ,

where we used2R_α[μν]β = - R_αβμνwhich follows from the Bianchi identity for the Riemann curvature.
These are Eqs. (<ref>)- (<ref>) in the main text.



§ EQUILIBRIUM DISTRIBUTION

In this Appendix, we derive the general form of the equilibrium distribution in the absence of the electromagnetic field, that is,A_μ= 0. The kinetic equation atO(ħ)reads

    δ(p^2)
     	[
     		p· D + ħ(D_μΣ_n^μν)D_ν
     		- ħ/2Σ_n^μνp^λ R_λρμν∂_p^ρ] f = 0  .

At equilibrium, the distributionfshould be a function of the
linear combination of the collisional conserved quantities, namely, the particle number, the linear momentum, and
the angular momentum. Therefore, we parametrize the equilibrium distribution as

    f = (g) ,  
     g =α(x) + β(x)· p + ħγ^μν(x)Σ_μν^n  ,

withα(x)=α_0(x)+ħα_1(x),β^μ(x)=β^μ_0(x)+ħβ^μ_1(x), andγ^μν(x)=γ^μν_0(x)+ħγ^μν_1(x)andO(ħ^2)terms omitted. Note thatn_μγ^μν=n_νγ^μν=0. Then the kinetic equation yields

    δ(p^2)
     (
     	p·∇α_0 + p^μ p^ν∇_μβ_0ν) = 0  ,

and

    δ(p^2)
     [p·∇α_1 + p^μ p^ν∇_μβ_1ν+
     	(D_μΣ_n^μν)(∇_να_0 + p^λ∇_νβ_0λ)
           
    	+ p· D (γ_0^μνΣ_μν^n)
    	- 1/2Σ_n^μν R_λρμνp^λβ_0^ρ]= 0  .

From Eq. (<ref>), we find

    ∇_μα_0 = 0  ,  ∇_μβ_0ν + ∇_νβ_0μ = ϕ_0(x) g_μν ,

whereϕ_0is an arbitrary scalar function.
Combined these constraints, the third term in Eq. (<ref>) is calculated as

    δ(p^2)(D_μΣ_n^μν)(∇_να_0 + p^λ∇_νβ_0λ) 
        = δ(p^2)
      		[
      			D_μ(Σ_n^μ[ν p^λ]∇_νβ_0λ)
      	  	- Σ_n^μν p^λ∇_μ∇_[νβ_0λ]] 
        = δ(p^2)
      		[
      			-1/2p· D
      				(Σ_n^μν∇_μβ_0ν)
    			+ 1/2Σ_n^μν R_λρμνp^λβ_0^ρ]  ,

where we have used p_με_νρσλ + p_νε_ρσλμ + p_ρε_σλμν + p_σε_λμνρ + p_λε_μνρσ = 0 ,∇_μ∇_[νβ_0λ] = -β_0^αR_αμνλ - g_μ[ν∇_λ] ϕ_0,2R_α[μν]β = - R_αβμν, andε^αμνρR_βμνρ=0.
Therefore, Eq. (<ref>) is reduced to

    δ(p^2)
      		[
      			p·∇α_1 
      			+ p^μ p^ν∇_μβ_1ν
      			+ p· D ( Σ_n^μνM_μν )		
      		] = 0 ,

whereM_=γ_0μν-1/2 ∇^⊥_[μβ_0ν]with⊥here representing the component perpendicular ton^μand we have used the fact thatΣ_n^μν∇_μβ_0ν = Σ_n^μν∇^⊥_μβ_0ν. Therefore, we have

    p·∇α_1 
     	+ p^μ p^ν∇_μβ_1ν
        + p· D(Σ_n^μνM_μν) 
      = ∑_l=1^∞ϕ_l(x)/2(p^2)^l ,

whereϕ_lwithl≥1are arbitrary functions. Making a scale transformation,p_μ→λp_μ, and comparing different powers ofλ, we obtain

    ∇_μβ_1ν + ∇_νβ_1μ = ϕ_1(x) g_μν,
       ϕ_l =0,      l≥2, 
        p·∇α_1 + p· D(
    Σ_n^μνM_μν)=0 .

To proceed, we decomposep_μwith respect ton_μas follows,p_μ=p_∥n_μ+p^⊥_μ. Substituting into Eq. (<ref>) we find that the following conditions

    ∇_μα_1=0,   M_μν=0

fulfill Eq. (<ref>) for arbitraryp^⊥_μ. However, in case thatn_μis a constant,∇_μn_ν= 0, the conditions to fulfill Eq. (<ref>) is

    n·∇α_1=0,  ∇_λ^⊥ M_μν=0 ,
       ∇^⊥_α_1 +1/2ε_ρσμνn^ν n·∇ M^ρσ=0  ,

which nevertheless contains constraint (<ref>) as a special case. Collecting Eq. (<ref>), Eq. (<ref>), and Eq. (<ref>), we obtain the equilibrium condition as given in  Eq. (<ref>) and Eq. (<ref>) in the main text up toO(ħ). We also notice that in the case of∇_μn_ν=0, the equilibrium state can maintain a difference betweenγ_0μνand the thermal vorticity1/2 ∇^⊥_[μβ_0ν]if a finite gradient of the chemical potential is present, as shown in the third equation in Eq. (<ref>), whose physical consequence deserves a more careful exploration in future.