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Chinese Physics C

Exploring the entanglement of free spin- , spin-1 and spin-2 fields *

Zhi Hui Yang, Ling-Yan Hung

Published Apr 25, 2019• 0 citations
Publisher

Abstract

Abstract In this study, we explore the entanglement of free spin- <?CDATA $ \displaystyle\frac{1}{2} $?> , spin-1, and spin-2 fields. We start with an example involving Majorana fields in 1+1 and 2+1 dimensions. Subsequently, we perform the Bogoliubov transformation and express the vacuum state with a particle pair state in the configuration space, which is used to calculate the entropy. This clearly demonstrates that the entanglement entropy originates from the particles across the boundary. Finally, we generalize this method to free spin-1 and spin-2 fields. These higher free massless spin fields have well-known complications owing to gauge redundancy. We deal with the redundancy by gauge-fixing in the light-cone gauge. We show that this gauge provides a natural tensor product structure in the Hilbert space, while surrendering explicit Lorentz invariance. We also use the Bogoliubov transformation to calculate the entropy. The area law emerges naturally by this method.

DOI
10.1088/1674-1137/43/5/053102

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Abstract In this study, we explore the entanglement of free spin- <?CDATA $ \displaystyle\frac{1}{2} $?> , spin-1, and spin-2 fields. We start with an example involving Majorana fields in 1+1 and 2+1 dimensions. Subsequently, we perform the Bogoliubov transformation and express the vacuum state with a particle pair state in the configuration space, which is used to calculate the entropy. This clearly demonstrates that the entanglement entropy originates from the particles across the boundary. Finally, we generalize this method to free spin-1 and spin-2 fields. These higher free massless spin fields have well-known complications owing to gauge redundancy. We deal with the redundancy by gauge-fixing in the light-cone gauge. We show that this gauge provides a natural tensor product structure in the Hilbert space, while surrendering explicit Lorentz invariance. We also use the Bogoliubov transformation to calculate the entropy. The area law emerges naturally by this method.
We compute the entanglement entropy of a massless spin 2 field in a sphere in flat Minkowski space. We describe the theory with a linearized metric perturbation field ${h}_{\ensuremath{\mu}\ensuremath{\nu}}$ and decompose the field into independent modes using tensor spherical harmonics. We fix the gauge such that (a) the two dynamical modes for each angular momentum decouple and have the dynamics of scalar spherical modes, and (b) the gauge-fixed field degrees of freedom inside the sphere represent gauge invariant operators of the theory localized in the same region. In this way the entanglement entropy turns out to be equivalent to the one of a pair of free massless scalars where the contributions of the l=0l=0l=0 and l=1l=1l=1 modes have been subtracted. The result for the coefficient of the universal logarithmic term is $\ensuremath{-}61/45$ and coincides with that computed using a regularization based on the mutual information.
Entanglement entropy has proven to be an extremely useful concept in quantum field theory. Gauge theories are of particular interest, but for these systems the entanglement entropy is not clearly defined because the physical Hilbert space does not factor as a tensor product according to regions of space. Here we review a definition of entanglement entropy that applies to abelian and nonabelian lattice gauge theories. This entanglement entropy is obtained by embedding the physical Hilbert space into a product of Hilbert spaces associated to regions with boundary. The latter Hilbert spaces include degrees of freedom on the entangling surface that transform like surface charges under the gauge symmetry. These degrees of freedom are shown to contribute to the entanglement entropy, and the form of this contribution is determined by the gauge symmetry. We test our definition using the example of two-dimensional Yang-Mills theory, and find that it agrees with the thermal entropy in de Sitter space, and with the results of the Euclidean replica trick. We discuss the possible implications of this result for more complicated gauge theories, including quantum gravity.
In gauge theories the presence of constraints can obstruct expressing the global Hilbert space as a tensor product of the Hilbert spaces corresponding to degrees of freedom localized in complementary regions. In algebraic terms, this is due to the presence of a center---a set of operators which commute with all others---in the gauge invariant operator algebra corresponding to a finite region. A unique entropy can be assigned to algebras with a center, giving a place to a local entropy in lattice gauge theories. However, ambiguities arise on the correspondence between algebras and regions. In particular, it is always possible to choose (in many different ways) local algebras with a trivial center, and hence a genuine entanglement entropy, for any region. These choices are in correspondence with maximal trees of links on the boundary, which can be interpreted as partial gauge fixings. This interpretation entails a gauge fixing dependence of the entanglement entropy. In the continuum limit, however, ambiguities in the entropy are given by terms local on the boundary of the region, in such a way relative entropy and mutual information are finite, universal, and gauge independent quantities.
Black holes in general relativity carry an entropy whose value is given by the Bekenstein-Hawking formula, but whose statistical origin remains obscure. Such horizons also possess an entanglement entropy, which has a clear statistical meaning but no a priori relation to the dynamics of gravity. For free minimally-coupled scalar and spinor fields, these two quantities are intimately related: the entanglement entropy is the one-loop correction to the black hole entropy due to renormalization of Newton's constant. For gauge fields, the entanglement entropy and the one-loop correction to the black hole entropy differ. This dissertation addresses two issues concerning the entanglement entropy of gauge fields, and its relation black hole entropy. First, for abelian gauge fields Kabat identified a negative divergent contribution to the black hole entropy that is not part of the entanglement entropy, known as a ``contact term''. We show that the contact term can be attributed to an ambiguous expression for the gauge field's contribution to the Wald entropy. Moreover, in two-dimensional de Sitter space, the contact term arises from an incorrect treatment of zero modes and is therefore unphysical. In a manifestly gauge-invariant reduced phase space quantization of two-dimensional gauge theory, the gauge field contribution to the entropy is positive, finite, and equal to the entanglement entropy. This suggests that the contact term in more than two dimensions may also be unphysical. Second, we consider lattice gauge theory and point out that the Hilbert space corresponding to a region of space includes edge states that transform nontrivially under gauge transformations. By decomposing these edge states in irreducible representations of the gauge group, the entanglement entropy of an arbitrary state is shown to be a sum of a bulk entropy and a boundary entropy associated to the edge states. This entropy formula agrees with the two-dimensional results from the reduced phase space quantization. These results are applied to several examples, including the ground state in the strong coupling expansion of Kogut and Susskind, and the entropy of the edge states is found to be the dominant contribution to the entanglement entropy.
Abstract In this paper, we explore the question of how different gauge choices in a gauge theory affect the tensor product structure of the Hilbert space in configuration space. In particular, we study the Coulomb gauge and observe that the naive gauge potential degrees of freedom cease to be local operators as soon as we impose the Dirac brackets. We construct new local set of operators and compute the entanglement entropy according to this algebra in 2 + 1 dimensions. We find that our proposal would lead to an entanglement entropy that behave very similar to a single scalar degree of freedom if we do not include further centers, but approaches that of a gauge field if we include non-trivial centers. We explore also the situation where the gauge field is Higgsed, and construct a local operator algebra that again requires some deformation. This should give us some insight into interpreting the entanglement entropy in generic gauge theories and perhaps also in gravitational theories.
In this paper, we explore the question of how different gauge choices in a gauge theory affect the tensor product structure of the Hilbert space in configuration space. In particular, we study the Coulomb gauge and observe that the naive gauge potential degrees of freedom cease to be local operators as soon as we impose the Dirac brackets. We construct new local set of operators and compute the entanglement entropy according to this algebra in 2+12+12+1 dimensions. We find that our proposal would lead to an entanglement entropy that behave very similar to a single scalar degree of freedom if we do not include further centers, but approaches that of a gauge field if we include non-trivial centers. We explore also the situation where the gauge field is Higgsed, and construct a local operator algebra that again requires some deformation. This should give us some insight into interpreting the entanglement entropy in generic gauge theories and perhaps also in gravitational theories.