Worldline approach for spinor fields in manifolds with boundaries

The worldline formalism is a useful scheme in Quantum Field Theory which has also become a powerful tool for numerical computations. It is based on the first quantisation of a point-particle whose transition amplitudes correspond to the heat-kernel of the operator of quantum fluctuations of the field theory. However, to study a quantum field theory in a bounded manifold one needs to restrict the path integration domain of the point-particle to a specific subset of worldlines enclosed by those boundaries. In the present article it is shown how to implement this restriction for the case of a spinor field in a two-dimensional curved half-plane under MIT bag boundary conditions, and compute the first few heat-kernel coefficients as a verification of the proposed construction. This construction admits several generalisations.


Introduction
The worldline formalism is a method to compute different quantities in Quantum Field Theory, such as effective actions, amplitudes, anomalies and partition functions.Unlike the traditional procedure involving Feynman diagrams, this formalism is characterised by the introduction of a point-particle whose dynamics is described using a first quantisation scheme.The transition amplitudes of this particle are used to compute the aforementioned quantities in the quantum field theory, which for unbounded manifolds can be computed using traditional quantum mechanical path integrals.Hence, first quantisation path integration can be formally used in this framework to extract information in quantum field theories.
For manifolds without boundaries, this worldline formalism is a well established and computationally efficient tool.Since the foundational works of Z. Bern and D. W. Kosower [1], and M. J. Strassler [2], it has been used for computing scattering amplitudes and effective actions for a variety of quantum field theories [3].Scalar, fermionic and vector fields have been established by Strassler in [2] for flat manifolds, and were later analysed in [4][5][6] for an arbitrary gravitational background.Higher-spin fields in conformally flat manifolds were addressed in [7][8][9].
In spite of the many models studied and applications considered, field theories in manifolds with boundaries have been more elusive.In order to construct a worldline formulation in these manifolds, one needs to restrict the path integration domain of the point-particle within its boundaries.Furthermore, this restriction also needs to take into account the specific boundary condition imposed on the field, which translates into imposing boundary conditions on the transition amplitudes of the point-particle.Therefore, the problem of formulating a worldline prescription for field theories in bounded manifolds reduces to the problem of constructing a path integral representation for the transition amplitudes in quantum mechanics.
Scalar fields in manifolds with boundaries have been extensively studied.For instance, Dirichlet boundary conditions on a (D − 1)-dimensional surface Σ can be modelled on the whole R D through the coupling λδ Σ (x) to a delta-function with support on Σ: in the limit of infinite coupling λ → ∞ one reproduces Dirichlet boundary conditions.This approach was introduced in the worldline context in [10] (for a similar mechanism for Neumann boundary conditions, see [11]).However, this procedure is generally not suitable for usual perturbative calculations: indeed, such procedures would lead to an expansion in positive powers of λ and the limit λ → ∞ usually appears to be ill-defined (for a strong coupling approach involving Padé approximants, see [12]).An exception occurs for the free scalar field in flat space, where a resummation that leads to the correct heat-trace expansion for the Dirichlet propagator in the limit of infinite coupling is possible [13] (for a similar resummation involving the Neumann propagator, see [14]).If instead of the free scalar field one considers a potential V (x), then the aforementioned resummation for the free Dirichlet propagator can be used to compute the contribution to the heat-trace at different powers of V (x), even for strong coupling [15].When the coupling constant is left finite, the delta-function coupling to the scalar field reproduces a semitransparent mirror, which was studied in the context of the worldline formalism in [16].
A rather different approach to the problem of a bounded field in the worldline formalism involves the use of image charges.This method was used in [17,18] for a scalar field with either Dirichlet or Neumann boundary conditions on the D-dimensional half-space M = R D−1 × R + limited by an infinite flat hyperplane.A generalization of the Neumann case to include Robin boundary conditions was later introduced in [19] for the same manifold.This technique strongly relies on the fact that the boundary is flat, so for other types of geometries an adaptation is necessary.In this context, the case of a scalar field confined to a D-dimensional ball B D (that is, the interior of the hypersphere S D−1 ) was analysed for Dirichlet and Neumann boundary conditions in [20], where a conformal transformation was performed to map the ball into the half-space.This same analysis was extended to include Robin boundary conditions in [21].
In the present work, a similar procedure in terms of image charges is performed for the case of a spinor field minimally coupled to an Abelian gauge field, confined inside certain smooth 2-dimensional curved manifolds M under MIT bag boundary conditions. 1 The procedure, which singles out in the path integral the contributions of worldlines which reach the boundary from those which lie entirely in the bulk, allows one to determine both diagonal and off-diagonal heat-kernel elements.The first ones correspond to closed worldlines within M that allow for the computation of the heat-trace asymptotics -and, thus, functional determinants-, while the second ones correspond to open worldlines that are useful for the computation of tree-level n-point functions within the context of the worldline formalism.Besides constructing a representation for the heat-kernel useful for either closed or open paths, the former scenario is considered in this article for the computation of the first heat-trace asymptotic coefficients (Seeley-DeWitt coefficients).
The reason for considering MIT bag boundary conditions is twofold.On the practical side, these local conditions imply that half of the spinor components satisfy a Dirichlet boundary condition while the other half satisfy a specific Robin boundary condition, and both types of conditions where previously analysed in the context of the worldline formalism for scalar fields.Other local conditions for fermions do not lead to a Robin boundary condition but instead to a more complicated first-order condition, such as the chiral bag boundary condition for which half of the spinor components satisfy a condition involving tangential derivatives, which have not so far been applied within a worldline framework.On the other hand, MIT bag boundary conditions are used in a vast number of physical situations: since they where introduced in [22] as part of an extended model for hadrons, they have also been applied to other contexts such as gauged supergravity [23], fermionic billiards [24], superstrings [25], spinning membranes [26] and graphene devices [27].It is expected that an analysis of MIT bag boundary conditions in terms of the worldline formalism will allow the implementation of worldline tools in these areas.
The procedure carried in the present article begins by representing the manifold M in coordinates that span the entire half-plane R × R + .In this coordinate representation, the manifold presents a certain metric g µν , and the boundary ∂M is represented via the straight line R.The next step is to duplicate M to build up another region M ≈ R 2 by reflecting the half-plane through its boundary and endowing the resulting full plane with the symmetric extension of the original metric.The curvature of the region M is different from the curvature of M because the symmetric extension introduces a Heaviside-function on the metric, which is thus non-smooth at the interface R. Besides, path integration of a point-like particle on curved space corresponds to a 0 + 1 sigma model which requires certain counterterms -specific to each regularization-that are necessary to maintain general coordinate invariance [28].In particular, the counterterm required by time-slicing renormalization contains a term proportional to the curvature of M , which is given by a delta-function with support at the interface.As a consequence, the computation of the heat-trace in these coordinates amounts to obtaining the point-particle expectation values of combinations of delta-and Heaviside-functions.Besides these curvature considerations, a symmetric extension of the tangential component of the gauge field in M is performed, as well as an antisymmetric extension of its normal component.The latter component is made continuous through a gauge fixing condition.
Image charges are used to separate "direct" and "indirect" contributions to the transition amplitude, according to whether the end-point of the trajectory lies in the physical region M ⊂ M or not.Finally, to illustrate the whole procedure, the leading direct and indirect contributions to the heat-trace are computed -which correspond to the volumes of M and its boundary ∂M -as well as the next-to-leading contribution to obtain the Seeley-DeWitt coefficient a 2 which gives the trace anomaly.A worldline formulation in phase space is employed for this application.
The organisation of the article is as follows.In Section 2 the relation between the heat-trace and the effective action for a quantum 1  2 -spin field coupled to an electromagnetic background is presented.The manifold M and the construction of the doubled manifold M , along with some geometrical properties, is described in Section 3.After this, geometrical and electromagnetic quantities in M must be "reflected" appropriately to M , which is done in Section 4. This completes the setup in terms of image quantities, from which path integration can be constructed in M .Since the field is fermionic, this path integration must take the Dirac gamma matrices into account.For this purpose, coherent states are chosen in the present article, so they are reviewed in Section 5. Next, the transition amplitudes for a fermionic point-particle in curved manifolds are introduced in Section 6, using path integrals in a phase space representation.This completes the setup for using first-quantisation path integrals to compute quantities in a fermionic Quantum Field Theory in manifolds with boundaries.This whole structure is then used considering a specific boundary condition: the MIT bag ones.For completeness, they are described briefly in Section 7. Then Section 8 contains the main result of this article, where an ansatz for a fermionic heat-kernel in M under MIT bag boundary conditions is presented.As an application of this construction, the first three Seeley-DeWitt coefficients of the heat-trace are computed in Section 9, showing perfect consistency with previous results.Finally, some considerations on the applications of this construction are drawn in Section 10, with particular emphasis on extensions to other boundary conditions for fermions and other kinds of fields, and also the implementations of the procedure to worldline numerical computations.

Effective action
Consider a spin− 1 2 field Ψ of mass m confined to a D−dimensional Euclidean manifold x ∈ M , minimally coupled to gravity and to an Abelian gauge field.The action reads where g is the determinant of the metric in M and2 .
is the (massless) Dirac operator.It contains an Abelian gauge connection A µ as well as a spin connection where e µ i and e i µ are vielbeins such that g µν = e i µ e j ν δ ij , δ ij = e i µ e j ν g µν and e µ i = g µν δ ij e j ν .The (flat) Dirac matrices γ i are constant and obey the Clifford algebra {γ i , γ j } = 2δ ij .In the present work an even-dimensional Euclidean manifold is considered, so Dirac matrices satisfy (γ i ) † = γ i and the chiral Dirac matrix γ ch = (−1) D/2 iγ 1 . . .γ D is well-defined.
The effective action up to one-loop order is (2.5) Using Schwinger's proper time regularization [29], the one-loop correction becomes which represents the divergent functional determinant in terms of the heat-trace of − / D 2 .
By means of the Schrödinger-Lichnerowicz formula [30,31], the Dirac operator squared can be written as a coordinate operator in the form where R is the scalar curvature of the manifold and In this form it is clear that − / D 2 is a Laplace-type operator with the fully Under quite general conditions, the heat-trace of − / D 2 admits the following asymptotic expansion: as µ, ν, ρ) denote curved indices in the bulk, while Latin letters from the middle of the alphabet (such as i, j, k) denote flat indices in the bulk.Also, Greek letters from the beginning of the alphabet (such as α, β, γ) denote curved indices in the boundary, while Latin letters from the beginning of the alphabet (such as a, b, c) denote flat indices in the boundary.For a D−dimensional manifold with boundaries, bulk indices range from 1 to D and boundary indices range from 1 to D − 1.
3 Here, "fully covariant" means that the Laplacian is written in terms of covariant derivatives with two types of connections: a curvature connection (the spin connection in this case) and also a gauge connection (which for the present article only includes the Abelian gauge potential).
Tr  [32].This is just one example of the applications of heat-kernel techniques to the perturbative study of quantum field theories.In this article it is shown how to compute the coefficients a n (− / D 2 , M ) for certain smooth 2-dimensional manifolds using worldline techniques.
3 Geometry of M and the 'doubled' manifold M In Section 6 it is shown that the heat-trace asymptotics (2.9) for the Laplace-type operator − / D 2 in M is determined by the path integral over closed trajectories of a point-particle.
In order to study the dynamics of this particle, it is convenient to identify M with a 2dimensional half-plane which is then embedded into a whole plane R 2 , denoted M , that represents two copies of the original manifold M glued together along the interface ∂M as in Figure 1.Hence, consider a smooth two-dimensional manifold representable in half-plane coordinates (x, y) such that −∞ < x < ∞ and 0 ≤ y < ∞.In particular, y = 0 represents the boundary ∂M .Let g µν be the metric in this coordinate representation, which will be considered as non-singular and diagonal, so in its matrix form one can write where both h ≡ h(x, y) and g ≡ g(x, y) are non-singular functions without zeros.Both of them have a straightforward geometrical interpretation: on one hand, when evaluated at y = 0, the function h(x, 0) is the metric of the boundary ∂M , so h(x, y) is an extension of h(x, 0) to the bulk of M .On the other hand, g = det g µν .Next, extend this metric to the whole plane R 2 by making a symmetric reflection with respect to the line y = 0.This "doubled manifold", which is denoted as M , has metric note that the doubled metric may no longer be analytic for it may have a discontinuous normal derivative at the fixed points y = 0.The corresponding integration measure is Denoting objects computed in M with a tilde " ", one can express the Christoffel symbols as where θ(y) is the Heaviside step function.In this and all subsequent geometrical expressions -unless something different is explicitly stated-, every object computed in M is evaluated at (x, y) and objects computed in M are evaluated at (x, |y|).The scalar curvature and the full contraction of two Christoffel symbols are note that the difference between the scalar curvatures of M and M has support at the boundary.
Path integration in curved manifolds requires the introduction of an additional counterterm potential to ensure coordinate invariance.In the present manuscript, phase space path integration will be used, which in curved spaces without boundaries is suitably described in terms of a Time Slicing formulation which involves the counterterm [28] ∆H Time Slicing construction for the computation of transition amplitudes is suitably defined in the manifold M , which has no boundaries.Therefore, the idea of the present article is to use path integrals in M and relate them to the transition amplitudes in M under MIT bag boundary conditions.Going back to M , one needs to consider the inward-pointing normal unit vector n µ at the boundary and extend it in some way to the interior of the manifold.From the metric (3.1) it follows As for the bulk of the manifold, a parallel transport extension along geodesics normal to the boundary will be considered.Therefore, n µ obeys in M the parallel transport equation The second fundamental form in ∂M has a single component L 11 and is given by Then the extrinsic curvature L, which is defined as the trace of the second fundamental form and is an invariant in ∂M , is which is an expression relating scalar invariants in M , ∂M and M .Going back to M , one needs to analyse the vielbeins e i µ that are present in the Dirac operator squared, both explicitly an implicitly through the spin connection ω µij .In the present article, the following choice will be used to set the vielbeins: at any point in M , let e i=2 µ be equal to the normal unit vector n µ .It follows that e i=1 µ must be equal to a unit vector perpendicular to n µ .This leads to a sign (directional) ambiguity that is solved by picking the direction that obeys the following 2-dimensional identity: At this point it must be remarked that with the aforementioned vielbein choice, every geometrical element in (2.7) must be non-singular.This ensures that the transition amplitudes to be constructed are well-defined in the entire manifold M .
Finally, to simplify expressions in two dimensions one could use the antisymmetry of the spin connection with respect to the flat indices to define ω µ ≡ ω µ12 ('1' and '2' are flat indices in this definition), which from (2.3) can be written as . This definition implies n µ ω µ = 0, as one can easily see using (3.9) and the vielbein choice discussed in the previous paragraph.In particular, this means that ω 2 | y=0 = 0.

Symmetry of functions and operators reflected to the lower half-plane
In the spirit of the method of images, one expects paths in the upper half-plane to have the same measure as reflected paths in the lower half-plane.To ensure this, several considerations about how to extend functions and operators must be taken into account.This task is carried out in the present Section.Firstly, the extension of geometrical quantities in M such as the metric g µν , the normal vector n µ , the Riemann covariant derivative ∇ µ and the spin connection ω µij is performed.Secondly, electromagnetic quantities such as the background gauge field A µ and the field tensor F µν are taken in consideration by analysing the geometry of gauge invariance.The results obtained lead to a natural extension of operators that depend on this geometrical and electromagnetic factors.

Reflection of geometrical quantities
If paths in the upper half-plane and their reflected paths in the lower half-plane must have the same value, then the line element ds = g µν x µ x ν must be the same in both half-planes.For this purpose, the metric g µν was extended symmetrically.In mathematical form and using (3.1), this reads where this extension means both functions have domain in M . 4To ensure that the lower half-plane is a physically equivalent reflection of the upper half-plane, every spinor in the theory is chosen to be reflected symmetrically as well: Next, consider the partial derivative operator ∂ µ acting on spinors.Since the spinors are reflected symmetrically, their partial derivative in the y-direction should be antisymmetrical: From metric (3.1) one can see that if an even (odd) number of indices in the Christoffel symbol are equal to 2, then the Christoffel symbol is proportional to a derivative in the x-direction (y-direction) of a metric component.Then, since the metric is reflected symmetrically, one gets if an even number of indices are equal to 2 , if an odd number of indices are equal to 2 , ( which leads to Reflections (4.4) also imply that the scalar curvature R and Γ 2 are reflected symmetrically.
Moving on to the normal unit vector n µ , the chosen extension consist in symmetrizing the component n 2 and antisymmetrizing the component n 1 , which has the advantage that the normal vector is not discontinuous at the interface y = 0.In mathematical form, Since n µ = e i=2 µ in the upper half-plane, the same vielbein choice can be set for the lower half-plane, and then e i=2 µ=1 (x, y) = −e i=2 µ=1 (x, −y) , e i=2 µ=2 (x, y) = e i=2 µ=2 (x, −y) .

Reflection of operators
Since the goal of the present manuscript is to express transition amplitudes in the manifold M (with boundary) in terms of transition amplitudes in the manifold M (without boundaries), then every operator Ô : H(M ) → H(M ) defined in the Hilbert space H(M ) must be somehow extended to a new operator Ô : H( M ) → H( M ) defined in the Hilbert space H( M ). 6This extension is written as where Ô> is the same operator as Ô : H(M ) → H(M ), and Ô< is computed by copying Ô> to the lower half-plane and introducing the reflection properties of Sections 4.1 and 4.2.This will ensure that any transition amplitude contained within the upper half-plane measure the same as the reflected transition amplitude contained within the lower half-plane.
To set ideas, consider the Dirac operator squared (2.7).Using (3.1) one can write, in two dimensions, Of course, this is defined in H(M ).The extension to H( M ) reads That is, (− / D 2 ) < is operationally similar to (− / D 2 ) > , which corresponds to the Dirac operator squared in the upper half-plane, except for the change of sign in the electromagnetic field tensor F and the spin connection ω µ .

Fermionic coherent states
If one aims to consider transition amplitudes of a point-particle on the manifold M , it is necessary to integrate not only on configuration space |x⟩ or phase space |x⟩ ⊗ |p⟩, but also on the internal degrees of freedom.For fermions, these degrees of freedom are given by the different spinor components.Two approaches exist for this purpose, namely the "Weyl symbol" method [33][34][35][36][37] and the "coherent state" method [38][39][40][41][42][43][44].The latter will be used in this manuscript and goes as follows. 7irstly, recall the extension (4.26) of the Dirac operator squared, written entirely in terms of the fermionic operators 1 and γ ch .Instead of representing operators acting on a spinor Ψ through these matrices, define the operators which obey the anticommutation relations {ψ, ψ † } = 1 and {ψ, ψ} = {ψ † , ψ † } = 0.Then, instead of describing the fermionic parts of Ψ through the spinor components Ψ 1 and Ψ 2 , one can construct the "vacuum state" |0⟩ and the "excited state" |1⟩ = ψ † |0⟩ such that the vacuum obeys ψ |0⟩ = 0 and ⟨0|0⟩ = 1.One can now define Both η and η are Grassmann numbers defined in such a way as to commute with the vacuum state |0⟩ and to anticommute with both operators ψ and ψ † .These states are coherent with respect to the operators ψ and ψ † : From these definitions, it can be proved that the trace of any zero-degree operator Ô ≡ O(ψ, ψ † ) is given by where Grassmann integration obeys Finally, the chiral Dirac matrix is identified with the operator where the subscript A stands for "antisymmetrized".
6 Transition amplitudes of a fermionic point-particle The transition amplitudes of a fermionic point-particle in a two-dimensional manifold are now at hand.Let H be the Hamiltonian of the particle.Now symmetrize it in terms of the bosonic operators x and p and antisymmetrize it in terms of the fermionic operators ψ and ψ † .This version of H, which is written as Hamiltonian of the particle, where V W is the corresponding effective potential, also Weyl ordered. 8The subscript S stands for "symmetrized".Consider the evolution of this particle at time T from state |x, y, η⟩, with fermionic coherent number η and located at position (x, y), to state ⟨x ′ , y ′ , η|.For convenience, the corresponding trajectories will be described as x(t) = x 0 (t) + q 1 (t) and y(t) = y 0 (t) + q 2 (t), where x 0 (t) and y 0 (t) consist on the straight line that connects the point x 0 (0) = x , y 0 (0) = y with x 0 (T ) = x ′ , y 0 (T ) = y ′ , and q 1 (t), q 2 (t) representing quantum fluctuations under homogeneous Dirichlet conditions q 1 (0) = q 2 (0) = q 1 (T ) = q 2 (T ) = 0. Similarly, paths in the coherent space will be described as η(t) = η + ψ(t) and η(t) = η + ψ † (t), where the quantum fluctuations obey ψ(0) = ψ † (T ) = 0. Then the transition amplitudes can be represented in terms of a bosonic path integral in phase space and a fermionic path integral in coherent-space [28]: 2) 8 An arbitrary potential will be considered in the present Section.
In the integrand, both g µν and V W are evaluated at x 0 (t) + q 1 (t) and y 0 (t) + q 2 (t).The effective potential V W is also evaluated at η + ψ(t) and η + ψ † (t).Evaluating the inverse metric at a fixed point (x, y), which in the present manuscript will be taken as the initial point, path integrals turn out to be normalised according to ) In order to keep track of the different powers of the (small) variable T , it is useful to turn to dimensionless quantities: t → T t, p µ (t) → p µ (t)/ √ T and q µ (t) → √ T q µ (t).Next, it is convenient to expand g µν in the kinetic factor around the initial point (x, y) as where ∆g µν is meant to be expanded in a power series around (x, y).It is also convenient to shift the momentum variables in the exponent and complete squares to get rid of the factors p 1 (x ′ − x) and p 2 (y − y ′ ).Defining e −ξ 2 /4T e ηη × DpDqDψ † Dψ e − 1 0 dt g µν (x,y) pµpν +∆g µν πµπν −ipµ qµ +ψ † ψ+T V W , ( where ∆g µν is evaluated according to (6.5) and V W is evaluated at π µ , x+t(x ′ −x)+ √ T q 1 (t), y + t(y ′ − y) + √ T q 2 (t), η + ψ(t) and η + ψ † (t).In order to turn (6.6) into a computational efficient expression, it is useful to define the expectation value of any arbitrary functional of the fields as The prefactor is chosen to fix the normalization according to ⟨1⟩ = 1.Then (6.6) turns into From the bosonic side, the expectation value in (6.8) represents the phase space integration over trajectories which satisfy homogeneous Dirichlet conditions q µ (0) = q µ (1) = 0 in configuration space and no restriction at all in momentum space.This makes the relevant quadratic operator in the Gaussian measure of (6.8) invertible.From the fermionic side, this expectation value represents the coherent-space integration over trajectories in terms of the "coherent fields" ψ(t) and ψ † (t) which satisfy ψ(0) = ψ † (1) = 0. Finally, in order to compute the expectation values it is convenient to define the generating functional as where j µ , k µ , κ and κ are "six" source fields -of which κ and κ are Grassmann fields.
Completing squares one finds with the Green functions given by 9 where ϵ(t − t ′ ) is the sign of t − t ′ with the convention ϵ(0) = 0. Any expectation value of a functional that is polynomial in the integrated fields can be computed as functional derivatives of the generating functional in terms of the source fields, with the successive evaluation at zero source.Other sources give the expectation value for other specific functionals (for example, see [20] for the computation of the expectation value of an exponential).In perturbative calculations, the relevant expectation values can usually be computed using Wick Theorem, for which only the two-point functions are needed.In the present case: 9 While G and • G are different components of the Green matrix, the notation used -following [28]clarifies that • G is equal to the derivative of G with respect to the first argument.If it had coincided with the derivative with respect to the second argument, the notation G • would have been used.

MIT bag boundary conditions
In this section, a brief introduction to (Euclidean) MIT bag boundary conditions in a two-dimensional manifold M is provided in terms of Dirac matrices as well as in terms of coherent states.Following Luckock [45], consider the projector whose rank is equal to one-half of the total number of spinor components.For D = 2, the rank is then equal to one.The MIT bag boundary conditions are given by These local conditions are sufficient to ensure the self-adjointness of the Dirac operator −i / D, which is a first-order differential operator.Equation (7.2) imposes Dirichlet boundary conditions on one-half of the spinor components.
When working with the second-order differential operator − / D 2 , an additional consideration is made as usual: the relevant functional space is spanned by the eigenfunctions of the Dirac operator −i / D. Therefore, not only does Ψ obey the MIT bag boundary condition but so does / DΨ, implying Π − / DΨ| ∂M = 0. Commuting / D with Π − one finds where L is the extrinsic curvature and is the projector complementary to Π − .Hence, while the Π − projection of a spinor satisfies the "Dirichlet-like" boundary condition (7.2), the complementary Π + projection satisfies the "Robin-like" boundary condition (7.3).When working with the Laplace-type operator − / D 2 , both (7.2) and (7.3) are used, leading to a set of mixed boundary conditions.The total number of these boundary conditions now becomes exactly equal to the total number of spinor components.
The same approach can be repeated in terms of operators acting on coherent states rather than using Dirac matrices.Focusing on the manifold M represented as a half-plane (whose boundary is the straight line y = 0) and recalling the vielbein convention established in Section 3, one finds / n = g µν e µ i n ν γ i = γ 2 .Thus, defining the difference between the complementary projectors as χ, it follows Then the mixed MIT bag boundary conditions (7.2) and ( 7.3) become ) 8 Heat-kernel in the manifold M This section summarises the main result of this article: a worldline representation for the heat-kernel of the Laplace-type operator − / D 2 on the manifold M expressed as a curved half-plane under MIT bag boundary conditions.Transition amplitudes (6.8) in the manifold M without boundaries are used for this purpose.Let be the heat-kernel on M , (i.e. the transition amplitudes of a fermionic point-particle evolving with Hamiltonian − / D 2 ).The ansatz proposed in the present article is where the terms in the RHS represent transition amplitudes of a point-particle in M evolving with Hamiltonian In this Hamiltonian, − / D 2 > is given by (4.27) and represents the Dirac operator squared − / D 2 as in H(M ), but symmetrized by being evaluated at (x ′ , |y ′ |).Note that since the RHS of (8.2) is defined in the entire double manifold M , then the LHS is defined in M as well.Therefore, the heat-kernel in the LHS of (8.2) is, formally, an extension of the heat-kernel in the LHS of (8.1) from M to M .The first (second) term in the RHS of (8.2) is referred as a 'direct' ('indirect') contribution to the heat-kernel.
To make sense of the ansatz (8.2), first is necessary to specify what is meant when, for an arbitrary operator Ô defined as in (4.23), one writes ÔK M .It specifically means that the operator is acting on the final point of the heat-kernel: For instance, if one considers the projectors Π ± (which satisfy Π > ± = Π < ± = Π ± ) then This results are a direct consequence of the factor χ in the last term of (8.2).Hence, the Π − (Π + ) projection of the heat-kernel is antisymmetric (symmetric) in terms of a reflection with respect to the boundary.This is a desired property of K M , because previous research for scalar fields involving Dirichlet, Neumann and Robin boundary conditions [17][18][19][20][21] has shown that if Dirichlet (Robin) boundary conditions are satisfied, then the heat-kernel is antisymmetric (symmetric).Equation ( 8.3) is also inspired by these results for scalar fields: in the literature, the original Hamiltonian in H(M ) has always been reflected symmetrically to the lower half-plane and, if Robin boundary conditions where obeyed, an additional delta term was introduced.The Π + projector accompanying the delta term in (8.3) ensures that this delta acts only on the projection that obeys a Robin boundary condition.
To prove the heat equation (8.7), expression (8.4) is used with Ô = − / D 2 , which is given by (4.26).Then, considering the relation properties of the Dirac matrices, one gets On the other hand, (8.12) From (8.11) and (8.12) it follows Although (8.13) is valid in the entire manifold M , note that its RHS is exactly zero in the bulk (where y ′ > 0), from which (8.7) is obtained.The final requirement to prove is the Robin condition (8.9).Firstly, consider the integral of (8.13), multiplied by √ g, with respect to y ′ from 0 − to 0 + : (8.14) Next, note from equation (8.5) that Π − K M is not necessarily continuous at the interface y ′ = 0, but as long as the discontinuity is finite, the integral of K M is zero.The same conclusion is drawn for any derivative of K M with respect to any variable other than y ′ .Those observations hold even if K M or its derivatives are multiplied by any function with finite discontinuity.Therefore, (8.14) becomes (8.15)This integral is solved easily if one considers the fact that ω 2 | y ′ =0 = 0 (see Section 3), the gauge semi-fixing condition (4.20) and the symmetry property (8.6): The Robin condition (8.9) follows trivially from (8.16) and (3.8).

Application: Seeley-DeWitt coefficients
As an application of the previous construction, the first three Seeley-DeWitt coefficients a 0 , a 1 and a 2 for the asymptotic heat-trace (2.9) are computed in the present Section.In two dimensions the coefficient a 2 represents the (integrated) conformal anomaly of the theory.The computation will be performed in phase space to avoid the introduction of ghost fields.Therefore, a phase space representation of the point-particle Hamiltonian (8.3) is needed.From the usual identification ∂ µ = ig 1/4 pµ g −1/4 , introducing the counterterms (3.7), and taking Section 5 into consideration to work in terms of coherent states instead of Dirac matrices, one finds the Weyl ordered expression where the subscript S stands for "symmetrized" with respect to the bosonic operators pµ and xµ .The heat-trace then admits the following expression: with the 'direct trace' Tr K dir M and the 'indirect trace' Tr K ind M given by The computation for each contribution will be carried out in two steps: first, a general non-perturbative expression will be obtained by solving the Grassmann trace.Then, a perturbative calculation up to order T 0 will provide the SDW coefficients a 0 , a 1 and a 2 .

Direct trace
From Sections 6 and 8 it is now known that one can write (9.3) as where the exponent in the mean value is given by Every function of the coordinates in (9.6) is evaluated at x + √ T q 1 , y + √ T q 2 unless explicitly stated.Note how H dir is of order √ T at least, which eases the perturbative expansion in powers of T .Expanding e − 1 0 dt H dir in powers of η and η and then solving the Grassmannian trace yields with In expressions (9.8)-(9.12)every function of the coordinates is again evaluated at x + √ T q 1 , y + √ T q 2 unless the evaluation point is explicitly stated.Equation (9.7) gives a non-perturbative result for the direct contribution of the heat-trace.
To compute the heat-trace up to any desired Seeley-DeWitt coefficient a k , now one expands (9.7) in terms of T up to order T −1+k/2 .Noticing that H dir is of order √ T at least, this means that one needs to expand the exponential containing this factor up to order k in the power series, thus "lowering" H dir up to k times in total.Then, defining one can see that In particular, computing up to the coefficient a 2 means expanding up to order T 0 , thus "lowering" H dir up to 2 times in total.Hence, one can readily write , one first expands up to order T 0 obtaining where b dir is expanded up to order T 1/2 , and in the product θ dir θdir one only retains the lowest order which is T 0 .In this scenario, using the mean values obtained at the end of Section 6, one gets with the RHS of both equations evaluated at (x, |y|).The integral of the Green function K(t, t ′ ) yields 1/2, so replacing into one gets and expanding up to order T 0 one obtains Focusing first on the factor 1 in the parenthesis, the mean value ⟨H dir (t 1 )⟩ needs to be computed up to order T .To start this calculation, consider the "δ-term" first: This expectation value can be computed by expressing the delta distribution in its Fourier representation: remains to be computed.Its expansion up to order T 0 is Next, one needs to expand the product H dir (t 1 )H dir (t 2 ) up to order T , which is simply the lowest order.Through similar calculations as before: Adding equations (9.16), (9.21) and (9.22), and after some geometrical identities, yields ) .(9.23)

Indirect trace
The indirect trace is a bit more involved than the direct one for several reasons.First, note that the transition amplitude in (9.4)This removes the operator χ in the transition amplitude at the expense of introducing the factor in parenthesis in the previous equation.Without χ, the transition amplitude to be integrated is ⟨x, −y, η | e −T H |x, y, η⟩, which can be expressed in terms of (6.8).The next complication is that now there is a multiplicative factor which is equal to e −y 2 g/hT .This term, when expanded as a power series, provides negative powers of T .To get rid of them, one can perform the rescaling y → √ T y.With all these ingredients, equation (9.4) where π 1 = p 1 and π 2 = p 2 − i y g(x, √ T y)/h(x, √ T y).Every function of the coordinates in (9.24) is evaluated at x + √ T q 1 , √ T (1 − 2t)y + q 2 unless explicitly stated.H ind is of order √ T at least, just like H dir .Expanding e − 1 0 dt H ind in powers of η and η and then solving the Grassmannian trace yields

25) with
) Once again, in expressions (9.26)-(9.29)every function of the coordinates is evaluated at x + √ T q 1 , √ T (1 − 2t)y + q 2 unless the evaluation point is explicitly stated.Equation (9.25) gives a non-perturbative result for the indirect contribution of the heat-trace.
One interesting result that follows from equation (9.25) is the fact that for boundaries ∂M obeying L = 0 one finds Tr K ind M = 0. Indeed, if L = 0 then ∆ ind = 0 and the mean value in the integrand becomes e − 1 0 dt H ind ( θind − θ ind ) .Since θind − θ ind is linear in the fields ψ and ψ † , it turns out that only those terms of e − 1 0 dt H ind that are also linear in the fields ψ and ψ † could give a non-zero mean value.But such terms do not exist if L = 0, proving that Tr K ind M = 0 if the boundary has zero extrinsic curvature.Another interesting fact lies in the aforementioned rescaling y → √ T y.Due to it, the evaluation point (x, y) of any function is replaced by (x, √ T y).Hence, the coefficients in the subsequent expansion in powers of T correspond to functions evaluated at (x, 0), that is, at the boundary.Therefore, indirect terms contribute only to boundary terms in the heat-trace expansion.
Note now that the RHS of equation (9.25) is of order T 0 at least, which implies that the indirect terms do not contribute to the Seeley-DeWitt coefficients a 0 and a 1 .For the coefficients a k with k ≥ 2, one expands (9.25) in terms of T up to order T −1+k/2 .Note that H ind is of order √ T at least, which means that one needs to expand the exponential containing this factor up to order k − 2 in its power series, thus "lowering" H ind up to k − 2 times in total.In particular, computing up to the coefficient a 2 means that one can completely ignore the exponential containing the factor H ind .Therefore, The mean value of the delta is solved analogously as in the direct case.Thus one gets Tr ) .(9.30)

The trace anomaly
Adding equations (9.23) and (9.30) leads to the heat-trace on the manifold M : By comparison with (2.9), the first three Seeley-DeWitt coefficients are obtained: These results are in perfect agreement with [32,46].In particular, the coefficient a 2 leads to the integrated trace anomaly: .31)In this article, an approach to consistently implement the worldline formalism (WLF) for spinor particles in manifolds with boundaries was introduced.The boundary condition under consideration was the MIT bag boundary condition.To better fix ideas, the construction was carried out in an arbitrary two-dimensional manifold M representable in half-plane coordinates (x, y), such that the bulk of the manifold consists on y > 0 and the boundary ∂M is the straight line y = 0.The construction relies on the method of images: the manifold was mirrored symmetrically with respect to the boundary, generating a new two-dimensional manifold M without boundaries.The old boundary became an interface were a singular curvature developed.To analyse the applicability of the method, the first three Seeley-DeWitt coefficients of the heat-trace were computed.In this approach, all curvature terms must be non-singular in the (x, y) coordinate representation of the manifold.
The procedure that has been set up admits generalisations and concrete applications.Up to now, worldline formulations on quantum fields with boundaries have been established only for scalar fields: firstly, Dirichlet and Neumann boundary conditions were analysed in flat space using the method of images [17], and Robin boundary conditions were considered later using the same method [18].Some years later, the method was extended to incorporate curved manifolds and curved boundaries, including compact manifolds [20,21].A different approach involving delta potentials, also for scalar fields, was analysed as well: firstly, a free scalar field with semitransparent Dirichlet boundary conditions was studied [13,16], and semitransparent Neumann boundary conditions were analysed more recently [14].Shortly after, the case of semitransparent Dirichlet boundary conditions for scalar fields with a potential was studied using the same method [15].None of the aforementioned worldline approaches considered, until now, spinor fields (but see [47,48] for previous path integral approaches for Dirac particles in the presence of delta-like potentials).The procedure described in the present article is expected to open the path to include new scenarios, including new fields and new boundary conditions, in the context of the WLF.
For the case of new fields, the approach followed in the present article, as described above, could be regarded as splitting the projections of the field components according to whether they obey Dirichlet or Robin boundary conditions.The same procedure should be valid for any field obeying the same mixed boundary conditions regardless of the Lorentzian nature of the field.For instance, a multiplet of several scalar fields or a gauge (vector) field are scenarios where the present approach is expected to hold.For the latter case, physically meaningful mixed boundary conditions are present in the literature [49][50][51][52] (see also Section 3.4 in [32] for a brief summary).The implementation of the procedure described in the present article to gauge fields with these conditions is currently under consideration.
For the case of other boundary conditions, it is well known that spinors could satisfy a large variety of them.In the area of local boundary conditions, MIT bag conditions are just one of many possibilities.Another closely related, physically meaningful possibility are the chiral bag boundary conditions, first introduced in [53] to construct a chirally symmetric theory in Quantum Chromodynamics.Since then, they were applied in other areas such as fermionic monopoles [54], fermionic billiards [24], graphene devices [55] and Weyl semimetals [56].These conditions can be written as Π − Ψ| ∂M = 0 using the non-Hermitian projector Π − = 1 2 1−i/ nγ ch e r(x α )γ ch , where r(x α ) is a real-valued function of the tangential coordinates.The main difficulty with these conditions is that, for r(x α ) ̸ = 0, the first-order induced boundary condition contains partial derivatives [57].Furthermore, firstorder boundary conditions with tangential derivatives are completely unexplored within the worldline frame.It would be interesting to see if one could implement a worldline procedure, similar to the one of the present article, to include these boundary conditions.Besides, they are reduced to MIT bag boundary conditions for r(x α ) = 0, so the present article could be use to check the construction of any possible procedure.
In the area of global boundary conditions, the Atiyah-Patodi-Singer (APS) conditions, sometimes referred as spectral boundary conditions, are well known in the literature [58,59].They are connected to index theorems [60][61][62][63] and raised interest in the study of topological devices [64].The Seeley-DeWitt coefficients of the heat-kernel asymptotics for these boundary conditions have been studied in the past [65,66].They can be written in terms of projectors Π ± , but they are usually expressed in terms of eigenvector components which are not perfectly suited for worldline computations.It would be interesting to see if APS boundary conditions can be implemented within a worldline perspective, where the familiar calculational efficiency of worldline techniques is expected to hold.
Besides the aforementioned generalisations, concrete applications are also at hand.On the one hand, fermions in manifolds with boundaries have been used to study quantities in Quantum Field Theories such as anomalies [67] and the Casimir effect [68].These approaches are based on delta-like potentials instead of the method of images.On the other hand, worldline numerical computations have been employed on the same areas [10,12,[69][70][71], mostly considering rigid boundaries and scalar fields.The approach provided in the present article could be used to study these phenomena analytically for spinor fields, as well to test numerical computations in manifolds with rigid boundaries.which appears frequently in the calculation of the direct trace.M represents any monomial of the fields q µ and p µ .In the following it will be shown how to compute (A.1) based on the ideas set in [20].
To solve the mean value with the step function, perform the Fourier transformation Then (A.1) becomes 2) The computation of the mean values in the RHS of (A.2) can now be carried out by expanding f and f A as power series with respect to the fields q µ .The mean value with the exponential is computed employing the generating functional given by (6.9) and (6.10) with j µ = −k δ 2 µ δ(t − t ′ ), where t ′ is the time at which the field q 2 (t ′ ) in the exponent is evaluated.An explicit list of several mean values involving this exponential is given in [20].
In calculations of the heat-trace, equation (A.2) appears integrated in the manifold M with measure √ g.In this scenario, introducing the rescaling y → √ T y yields where the result of the mean value in the third line should be evaluated at (x, √ T y) due to the rescaling.Note that the last term (second and third line) in the RHS is at least of order T : indeed, the lowest explicit order is √ T , but its coefficient is proportional to f A (x, 0) = 0.
Several particular applications of equation (A.3) are of interest for the calculations of Section 9.As an example, consider the monomial M = p µ p ν and f = g µν .Up to order T , equation (A.where each curvature element in the integral of the last line is being evaluated at (x, 0).As a final remark, the present procedure to compute mean values can be extended to the case of products of several functions ⟨f 1 f 2 . . .f l ⟩, where each function f i is being evaluated at x + √ T q 1 (t i ), y + √ T q 2 (t i ) .For each of them it is possible to write, as before, a "bulk term" plus a "θ-term", with its own Fourier transformation for each step function.

Figure 1 :
Figure 1: Worldlines in two dimensions.Picture 1a shows a worldline in the manifold M from initial point • (black) to final point • (green), hitting the boundary once in the intermediate point • (blue).Picture 1b represents M , where the boundary turns into an interface.Point • (pink) in the lower half-plane is the reflection of • (green) along this interface.A typical curve from • (black) to • (green) that hits the boundary once at • (blue) has a corresponding curve from • (black) to • (pink): the segment from • (black) to • (blue) remains the same, while the segment from • (blue) to • (green) gets symmetrically reflected from • (blue) to • (pink).The contribution of the original curve is called 'direct', and the contribution of the second one is called 'indirect'.Both curves measure equally.