Proof of the quantum null energy condition for free fermionic field theories

The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $\langle T_{kk}\rangle_{p}\geq lim_{A\rightarrow 0}\left(\frac{\hbar}{2\pi A}S_{out}^{\prime\prime}\right)$ where $S_{out}$ is the entanglement entropy restricted to one side of a codimension-2 surface $\Sigma$ which is deformed in the null direction about a neighborhood of point $p$ with area $A$. A proof of QNEC has been given before, which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and methods, we prove the QNEC for fermionic field theories.


I. INTRODUCTION
In general relativity, energy conditions are restrictions imposed on the energy-momentum tensor of matter and (nongravitational) fields to prevent unphysical solutions of Einstein's field equations. There are several energy conditions, of which the null energy condition (NEC) is one of the weakest. It states that where T ab is the energy-momentum tensor and k a is an arbitrary null vector. In spite of being weaker than other energy conditions, the NEC is sufficient to prove many important results, such as the Penrose singularity theorem [1], the second law of black hole thermodynamics [2], and other area laws [3] etc. It is well known, on the other hand, that the NEC and all other energy conditions are violated by quantum field theories, even free ones. So finding generalizations of these conditions when quantum fields are included is an important issue which could help generalize other results in general relativity that depend on classical energy conditions, provide insights into the nature of quantum gravity, and further highlight the connection between energy and information. For example, the vacuum modular Hamiltonian of a Rindler wedge is given by the boost generator for any quantum field theory [4]. The QNEC has been used to investigate modular Hamiltonian for more general half spaces [5].
The QNEC was conjectured by considering the covariant entropy bound [6][7][8], an entropy bound which attempts to reformulate and generalize the Bekenstein entropy bound [9,10] in a covariant way, and the related quantum focusing conjecture [11], a generalization of the focusing theorem for null geodesics. The QNEC states that where S out is the von Neumann entropy or entanglement entropy restricted to one side of a codimension-2 surface Σ with the derivatives taken with respect to deformations in the null direction about a neighborhood of point p with infinitesimal transverse area A. More details are provided in Sec. II. The QNEC has been proved for holographic theories and CFTs with a twist gap [12][13][14][15]. In this paper, we provide a new proof of the QNEC for free and super-renormalizable fermionic field theories using a similar method for the proof with bosonic field theories [16].

A. Overview
The QNEC has been proved for free and super-renormalizable bosonic field theories at any point that lies on a stationary null surface N, with Σ being an arbitrary cut and deformed along N [16] (see Fig. 1). The restriction to such points was necessary as the proof relied on quantization on a null surface (null quantization), which can only be applied to stationary null surfaces. Such null surfaces include the Rindler horizon in Minkowski space and the horizon of eternal Schwarzchild and Kerr black holes.
Null quantization allows the proof to reduce to just working with decoupled free left-moving chiral fermionic 1 þ 1 CFTs and an auxiliary system whose exact details are not needed. The state of the system was constructed via a path integral and various traces evaluated on replicated manifolds to calculate the Renyi entropies. The replica trick was used to calculate the von Neumann entropy via analytically continuing the Renyi entropies, which allowed the explicit evaluation of ℏ 2πA S 00 out − hT kk i to zeroth order in A. To prove the QNEC, it is sufficient to prove that the expression above is less than or equal to zero for small A. This will also be our strategy for proving the QNEC for free and super-renormalizable fermionic field theories.

II. THE QUANTUM NULL ENERGY CONDITION
If we choose a point p in a globally hyperbolic spacetime, a null vector k a , and any smooth codimension-2 surface Σ 1 that (i) Σ partitions a Cauchy surface into two, (ii) Its boundary ∂Σ contains the point p, and (iii) Σ is normal to k a with vanishing expansion at p, then we can consider deforming the surface Σ about a neighborhood of p, with a small area A along the null geodesics generated by k a . By choosing an affine parameter λ for the null generators, we can label the deformed surfaces as ΣðλÞ with Σð0Þ ¼ Σ. Then the QNEC states that where S out is the entanglement entropy on one side of ΣðλÞ and the derivative is taken with respect to λ. The choice of side can be done arbitrarily.
To be more explicit, we can set up a transverse coordinate system (y coordinate) for ∂Σ in a neighborhood of p and define null vectors k a ðyÞ normal to ∂Σ with k a ðpÞ ¼ k a . The vanishing expansion condition is with respect to this family of null vectors.
With our choice of affine parameter, we now have a coordinate system ðλ; yÞ with the location of ∂Σ at ð0; yÞ. Deformations of Σ now correspond to its boundary ∂Σ shifting along the λ coordinate about p. ΣðλÞ is located at ½Vðy; λÞ; y where Vðy; λÞ ¼ 0 everywhere except in a neighborhood of p with transverse area A for which Vðy; λÞ ¼ λ.
Our proof of the QNEC applies to free and superrenormalizable fermionic field theories for which p lies on a stationary null surface (vanishing null expansion everywhere) with a normal vector k a and Σ a section of this null surface, as in the bosonic case [16]. Without a loss of generality, we choose k a to be future directed and choose the side of Σ towards which k a points. Initially, we work in dimension D > 2 to ensure we have a transverse direction to deform Σ on.

III. NULL DISCRETIZATION AND QUANTIZATION
A stationary null surface N can be obtained as a limit of Cauchy surfaces and by unitary invariance; S out on the Cauchy surfaces and the null surface N with part of the future null infinity are equal. Hence, by using the null surface as an initial surface and quantizing on N (null quantization), we can calculate S out by restricting the state to the future of Σ. Null quantization requires N to be a stationary null surface [17,18].
We first discretize N along the transverse direction (y direction) into regions of small transverse area A called pencils. We take this transverse area to be the same as the area in Eq. (3) (i.e., the size of the neighborhood of p about which Σ is deformed) as we take the limit when A → 0. In this way, deformations of Σ are equivalent to deformations along the distinguished pencil (the pencil which contains p).
With this setup, it has been shown [16,17] that restricted to N, the theory decomposes into a product of 1 þ 1 dimensional free left-moving chiral conformal field theory (CFT), with K 2 CFTs associated to each pencil of N, where K is the number of components of the spinor field, and thus, the vacuum state factorizes with respect to this pencil decomposition of N. For small A, the state of the system can be written as [16] FIG. 1. The codimension-2 surface Σ splits some Cauchy surface into two disjoint regions. One side (yellow surface) is unitarily equivalent to the stationary null surface N (green surface) to the future of Σ with part of the future null infinity. Hence, S out on both these surfaces are equal. Deformations of Σ are equivalent to deformations on the distinguished pencil.
where ρ ð0Þ pen ðλÞ is the vacuum state on the distinguished pencil restricted to the future of ΣðλÞ, ρ ð0Þ aux is some state on all the other pencils on N, and part of future null infinity (auxiliary system) and σðλÞ is a small perturbation. For the proof of the QNEC, we only need to consider terms in σðλÞ up to order A 1 2 . Such terms are constructed by taking the partial trace of terms of the form j0ih1j and j1ih0j in the distinguished pencil Fock basis because they scale like A 1 2 . In general, jnihmj scales like A nþm 2 [16]. This will be partially reviewed in Sec. IV. From now on, we refer to the distinguished pencil simply as the pencil and all other pencils together as the auxiliary system.
When "restricted" to the full pencil, the state is near the vacuum state ρ ð0Þ pen for small A. Up to the order A 1 2 , the state on N must be of the following form 2 : where 3 jii and jji together form an orthonormal basis for the auxiliary system such that jiihjj and jjihij together forms a Grassmann-odd basis of operators. For example, jii could label states obtained by applying even creation operators to the vacuum state (a † 1 a † 2 j0i) and jji labels states obtained by applying odd creation operators to the vacuum state (a † 1 a † 2 a † 3 j0i). Without a loss of generality, we pick jii and jji that diagonalizes ρ ð0Þ aux . jψ ij i are single particles states in the CFT so that jψ ij ih0j and j0ihψ ij j takes the schematic form j0ih1j and j1ih0j and represents the perturbation σ of the state on N at order A 1 2 (see Sec. III). Higher order terms in A represented in "…" can be ignored.
The single particle states jψ ij i can be constructed applying the single-field operator on the vacuum state. In the Euclidean path integral picture, the most general single particle states can be constructed by insertions of the single-field in the Euclidean plane, as shown in Fig. 2. Since chiral fermionic fields only depend on the coordinate z ¼ x þ t ¼ x − iτ, translations along the Rindler horizon (null surface) are equivalent to translations along the spatial x direction [16].
To trace out the degrees of freedom over x < 0 and obtain the state of the system when λ ¼ 0, we insert a branch cut from the origin to x ¼ ∞ on the Euclidean plane. The matrix elements of the state are represented via the path integral picture schematically 4 as [19] with boundary conditions defined just above and below the branch cut such thatψðxÞjψi ¼ ψðxÞjψi. M is a normalizing constant, and O ρ is some operator that defines ρ. In our case, O ρ will be of the form I þ A 1 2Õ ρ , withÕ ρ constructed from single field insertions.
To obtain the state of the system for any λ, we need to take the partial trace along x < λ on the pencil. This can be done by moving the branch point from Alternatively, we can translate the field insertions to the left by λ. From this point of view, the vacuum state, FIG. 2. The state of the CFT can be constructed in the path integral picture with insertions of ψ. The state on the full pencil is obtained by moving the branch point to x ¼ −∞. The state for some value λ is obtained by moving the branch point to x ¼ λ.
2 For simplicity, we have assumed that we have one chiral CFT on the pencil. To include more CFTs, one can add the corresponding terms to Eq. (5). However, since we only need to consider the second order term of the entropy (see Sec. IV), we can consider the extra terms separately and simply sum over them to obtain S ð2Þ . 3 We have ensured that the state ρ is a physical state in the sense that the state is Hermitian and invariant under a 2π rotation. For example, if we consider the vacuum state j0i and the single particle state j1i for some fermionic quantum system, thenρ 1 ¼ j0ih0j þ j1ih1j is a physical state butρ 2 ¼ j0ih1j þ j0ih1j is an unphysical state since after a rotation by 2π,ρ 2 → −ρ 2 . In other words, a physical state must be a Grassmann-even operator. 4 For a fermionic system, the functional integral is in terms of Grassmann variables. Thus, when taking the trace in the functional integral, we have to sum over antidiagonal elements, and so the upper boundary condition has an extra minus sign. Additionally, one can not directly construct a path integral as in Eq. (6) for Majorana spinors without some modification. See the discussion in Sec. V for more details.
does not depend on λ [16]. The modular Hamiltonian K pen is also the Rindler boost generator [20] (up to an additive constant).
Using this, we can evaluate the trace of the state [Eq. where Recall that we chose a basis for the auxiliary system such that ρ ð0Þ aux is diagonal. Hence, K aux is also diagonal with an eigenvalue K aux jii ≔ K i jii. If we define f ii ðr; θÞ ¼ f jj ðr; θÞ ¼ 0 for all i and j, then we can rewrite Eq. (8) as where μ, ν are both summed over i and j indices. For definiteness, we define for all μ and ν, which agrees with the previous definition for matching i and j.
To ensure that the state (9) is Hermitian, we must impose the following reality condition on f μν ðr; θÞ: Other than this condition, f μν ðr; θÞ can chosen arbitrarily (see the Appendix for more details).
From now, we will write K ≔ K tot and and thus, Note: We can substitute OðλÞ forÕ p to define ρðλÞ in Eq. (6).

IV. EVALUATION OF S ð2Þ00
Our state [Eq. (9)] is in the form, We can expand the entanglement entropy perturbatively in σ, where S ðnÞ contains n powers of σ. We assume without loss of generality that ρ pen ⊗ ρ ð0Þ aux is normalized and TrðσÞ ¼ 0 so that ρðλÞ is also normalized.
It has been shown that under these conditions [11,16], where we define f 00 ¼ d 2 dλ 2 fðλÞj λ¼0 for any function f. Subtracting S 00 out from both sides of Eq. (16) and slightly rearranging gives [11,16] ℏ 2πA S 00 where terms in "…" contain terms higher than the quadratic order in σ or equivalently, higher than the zeroth order in A.
The QNEC states that the left-hand side of Eq. (17) is negative in the limit as A → 0. In this limit, only the first term ( ℏ 2πA S ð2Þ00 ) on the right-hand side is nonzero, and so to prove QNEC, it is sufficient to prove that S ð2Þ00 ≤ 0 for perturbations about the vacuum.
Note: In general, σ will contain terms proportional to A n 2 for n ≥ 1. However, only the terms proportional to A 1 2 are relevant for proving the QNEC as contributions of higher order terms will drop out as A → 0 in Eq. (17). Hence, we can ignore the "…" terms in Eq. (5).
As in the bosonic case [16], we will use the replica trick to calculate S ð2Þ00 . The replica trick is used to calculate entanglement entropies as [21] 5 To see this, consider angular quantization with an origin at ðt; xÞ ¼ ð0; 0Þ. Then, isolating the single field insertion, we have that hψjρjψ 0 i ∝ R ψðr;θ¼2π − Þ¼ψðrÞ ψðr;θ¼0 þ Þ¼ψ 0 ðrÞ ½dψψðre iθ Þe −S R E ∝ hψje −2πK pen ψðre iθ Þjψ 0 i. To obtain the most general form, ψ needs to smeared out on the Euclidean plane. S R E is the Euclidean action for Rindler coordinates [4].
whereZ n ¼ Tr½ρ n and D is an operator defined by for some function fðnÞ. Notice that to apply Eq. (19), we must analytically continueZ n to real n > 0.
It has been shown [16] that at the quadratic order in σ, logZ n can be written as where h…i n ≔ Tr½ðρ ð0Þ Þ n T½… Tr½ðρ ð0Þ Þ n : ð22Þ T½… is the θ ordering, and This is equivalent to the Heisenberg evolution of O in the angle θ by 2πk. We can obtain O ðkÞ from O by letting the range of integration that defines O shift from ½0; 2π to ½2πk; 2πðk þ 1Þ as long as we define f μν ðr; θÞ to be antiperiodic with period 2π 6 [16]. Hence, A. The correlators The same-sheet correlator, D −n 2 hOOi 00 n , has previously been evaluated [16] to give where TðxÞ ¼ −2πAT kk ðxÞ, TðxÞ being the energymomentum tensor of the CFT [17], and we write hÁ Á Ái ≔ hÁ Á Ái 1 . Explicitly, using Eq. (13), this can be written as where we Fourier expand f μν ðr; θÞ as Notice that the extra 1 2 term in the exponential automatically makes f μν ðr; θÞ antiperiodic as required. In Fourier components, the reality on f μν ðr; θÞ becomes where we define α μν ≔ K ν − K μ . For n ¼ 1, p ∈ Z so that after doing the angle integration, using the Kronecker delta from the integration, redefining the dummy variable m → m − 2 for the first term and m → m − 3 for the second term, we find that This can be further simplifed by redefining the dummy variables for the second term with The traces in Eq. (22) can be evaluated via a path integral on a n-replicated manifold, and so we interpret O ðkÞ as O inserted on the (k þ 1)th replica sheet. The n-replicated manifold is a Riemann manifold constructed by n copies of the Euclidean plane with various line segments identified (see Fig. 3). Thus, the sums over n and integrations over θ ∈ ½0; 2π can be replaced with one integration over θ ∈ ½0; 2πn [see Eq. (35)], as this covers the whole replicated manifold. ψ ¼ ψðr; θÞ is now a chiral left-moving fermionic field defined on the replicated manifold rather than the Euclidean plane. The definition of ψ for any angle is still given by the Heisenberg evolution rule from Eq. (A8) [16]. fðr; θÞ is still antiperiodic and defined via its Fourier expansion for all angles. Due to the different boundary conditions depending on whether n is even or odd, the correlation functions have to be evaluated separately for each case [19,22]. Instead, we will only focus on the case that n is odd and use this to analytically continue our expression to the positive real line. For now, we will assume that n is odd.
Hence, using the above, we have We show in the Appendix that, where q ∈ Z n þ 1 2n (i.e., q is summed over some odd integers divided by 2n) and Notice that when n ¼ 1, q is summed over just two values ( 1 2 and − 1 2 ) for which P ¼ 0. This is expected since for n ¼ 1, we have translation invariance, and so the CFT correlator should not depend on λ. This fact will be useful later.
Also, we show in the Appendix that where p ∈ Z n . Putting everything together into Eq. (36), we get since −p − m and p − m 0 are already in Z n . Focusing on −q − 5 2 , we rewrite q ¼ I n þ 1 2n where I ∈ Z. Thus, Since n ≥ 1 and odd, we can write n ¼ 2t þ 1 for integer t ≥ 0 so that A similar result holds for q − 3 2 . This means that doing the angle integration in Eq. (41) gives a Kronecker delta multiplied by 2πn. After relabeling m → m − 2, we have We have moved the operator D inside the integral and set n ¼ 1 for terms outside D. To do this, we used the fact that for any two functions fðnÞ and gðnÞ such that fð1Þ and d dn fðnÞj n¼1 are finite and gðnÞ ¼ 0, we have [16] DðfðnÞgðnÞÞ ¼ fð1ÞDgðnÞ: In Eq. (44), gðnÞ is the sum over q inside D, which we already showed above is zero for n ¼ 1. To apply the operator D, we must analytically continue gðnÞ. The method we use to analytically continue gðnÞ is similar to the method used for the bosonic case [16].

B. Evaluation of operator D
To analytically continue the expression inside D, notice that where in the first line, we removed the signðqÞ term by rewriting the sum over 0 < q < 1 or equivalently over 1 2n ≤ q < 1 − 1 2n . In the second line, we split the sum over q into two sum over 1 2n < q < 1 2 − 1 n and 1 2 þ 1 n < q < 1 − 1 n . Recall that PðAE 1 2 ; r 0 r Þ ¼ 0, and so we can remove the sum over q ¼ 1 2 . We have also defined z m ≔ −m − iα μν . By writing q ¼ k n , we can rewrite the sum in Eq. (46) as where k ∈ Z þ 1 2 .

First sum
Focusing on the first sum of Eq. (47), we need to find We can extend PðzÞ ≔ Pðz; r 0 r Þ to complex z, making PðzÞ an analytical function, independent of n. We can write Eq. (48) as [16] D X 1 2 ≤k≤ and evaluate the two terms separately. We can write where we obtain the second line by expanding PðzÞ about z ¼ − 1 2 as After using the following identity: we can write Eq. (50) as To evaluate D P 1 2 ≤k≤ n 2 −1 ð k n þ 1 2 Þ s , we write n as n ¼ 2t þ 1 for integer t and relabel k → k − 1 2 so that where now k ∈ Z in the second term. The expression inside D can now be analytically continued. Using the Hurwitz zeta function ζðs; aÞ, where we can write Eq. (54) as where to get to the second line, we applied Eq. (45). The Hurwitz zeta function is analytic in a which means that we can apply the operator D. Recall that D ¼ ð1 − n∂ n Þj n¼1 ¼ ð1 − ð2tþ1Þ 2 ∂ t Þj t¼0 and that the expression inside D is zero when t ¼ 0, so we can simplify Eq. (56) as Using ∂ a ζðs; aÞ ¼ −sζðs þ 1; aÞ, we find that Eq. (56) can be evaluated as − 1 2 ½sζð−s þ 1; 1Þ − 2sζð−s þ 1; 1Þ where ζðsÞ ¼ ζðs; 1Þ is the Riemann zeta function.
Putting everything together, we find that Similarly, for the Pð− k n Þ−Pðz m Þ k n þz m term from Eq. (49), we can write where we expanded PðzÞ ¼ P ∞ r¼0 a r ðz − 1 2 Þ r . The above can be written as and can be evaluated to give Now, we need to evaluate the last term of the first sum; i.e., we need to evaluate from Eq. (49), We can write this as [16] Pðz m ÞD where we have already relabeled k so that k ∈ Z. This expression can be written in terms of digamma functions, ψðzÞ, as where in terms of the Gamma function ΓðzÞ, However, we can not apply D straight away since we need to select the correct analytic continuation to real positive t. The digamma function has poles at zero and all negative integers, and so the expression in Eq. (64) may have poles for real positive t.
The method used to select the correct analytic continuation has already been done for the bosonic case [16], which avoids poles along real positive t and can straightforwardly be applied to our case with some minor modifications. One has to consider the three cases m < 0, m ¼ 0 and m > 0 separately, which leads to Notice that for s even, ð1 − ð−1Þ s Þ ¼ 0 and for s > 1 and odd, ζð1 − sÞ ¼ 0. Hence, only when s ¼ 1 does the sum above over s contribute in Eqs. (72b) and (72c). We can rewrite these sums over s as [16] ζð0Þ X ∞ r¼2 a r z m − 1 2 and ζð0Þ X ∞ r¼2 b r z m þ 1 2 Recall that P ¼ Pðz m ; r 0 r Þ given by Eq. (39). Hence, a 0 ¼ b 0 ¼ 0, a 1 ¼ 4ð r 0 r Þ