Abstract The Minkowski vacuum is often presented in textbooks and reviews as a thermofield double (TFD) state, an entangled state of field modes in the left and right Rindler wedges. This picture is widely used to explain the Unruh effect, motivate entanglement entropy calculations, and connect quantum field theory to black hole thermodynamics and AdS/CFT. However, we show that this interpretation, while elegant, is not exact. We explicitly compute two-point functions and their derivatives for a massless scalar field in two-dimensional Minkowski space, comparing results obtained from canonical quantization with those obtained by assuming a TFD form of the vacuum. Mixed-derivative correlators agree perfectly, but higher-derivative correlators show systematic mismatches that persist even for points well away from horizons and are not removed by infrared regularization. To further test this picture, we construct an alternate coordinate system that divides Minkowski spacetime into two disconnected regions, apply the same derivation that leads to the standard TFD expression, and obtain a new ‘entangled-state’ representation of the vacuum that is not thermal. This demonstrates that the appearance of a TFD structure is a feature of the derivation method rather than a fundamental property of the vacuum. Our results clarify the limits of interpreting the Minkowski vacuum as a literal TFD state, emphasizing that while it captures key thermal features, it should be viewed as a powerful calculational tool rather than a precise statement about Hilbert space structure.
The generators of radial conformal symmetries in Minkowski space-time can be put in correspondence with generators of time evolution in conformal quantum mechanics. Within this correspondence we show that in conformal quantum mechanics the state corresponding to the inertial vacuum for a conformally invariant field in Minkowski spacetime has the structure of a thermofield double. The latter is built from a bipartite ”vacuum state” corresponding to the ground state of the generators of hyperbolic time evolution. These can evolve states only within a portion of the time domain. When such generators correspond to conformal Killing vectors mapping a causal diamond in itself and generators of dilations, the temperature of the thermofield double reproduces, respectively, the diamond temperature and the Milne temperature found for massless fields in Minkowski spacetime. Moreover we calculate the entanglement entropy associated to the thermofield double states and obtain a UV divergent logarithmic behaviour analogous to known results in two-dimensional conformal field theory in which the entangling boundary is point-like.
Motivated by holographic complexity proposals as novel probes of black hole spacetimes, we explore circuit complexity for thermofield double (TFD) states in free scalar quantum field theories using the Nielsen approach. For TFD states at t = 0, we show that the complexity of formation is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals. For TFD states at t > 0, we demonstrate that the complexity evolves in time and saturates after a time of the order of the inverse temperature. The latter feature, which is in contrast with the results of holographic proposals, is due to the Gaussian nature of the TFD state of the free bosonic QFT. A novel technical aspect of our work is framing complexity calculations in the language of covariance matrices and the associated symplectic transformations, which provide a natural language for dealing with Gaussian states. Furthermore, for free QFTs in 1+1 dimension, we compare the dynamics of circuit complexity with the time dependence of the entanglement entropy for simple bipartitions of TFDs. We relate our results for the entanglement entropy to previous studies on non-equilibrium entanglement evolution following quenches. We also present a new analytic derivation of a logarithmic contribution due to the zero momentum mode in the limit of vanishing mass for a subsystem containing a single degree of freedom on each side of the TFD and argue why a similar logarithmic growth should be present for larger subsystems.
This paper studies the complexity between states in quantum field theory by introducing a Finsler structure based on ladder operators (the generalization of creation and annihilation operators). Two simple models are shown as examples to clarify the differences between complexity and other conceptions such as complexity of formation and entanglement entropy. When it is applied into thermofield double (TFD) states in
d-dimensional conformal field theory, results show that the complexity density between them and corresponding vacuum states are finite and proportional to ${T}^{d\ensuremath{-}1}$, where
T is the temperature of TFD state. Especially, a proof is given to show that fidelity susceptibility of a TFD state is equivalent to the complexity between it and corresponding vacuum state, which gives an explanation why they may share the same object in holographic duality. Some enlightenments to holographic conjectures of complexity are also discussed.
Consider a massless scalar field in Minkowski spacetime
M in its vacuum state and two Rindler wedges
R1 and
R2. Here
R2 is shifted to the right of
R1 by a distance $\mathrm{\ensuremath{\Delta}}$. We, therefore, have ${R}_{2}\ensuremath{\subset}{R}_{1}\ensuremath{\subset}\mathcal{M}$ with the symbol $\ensuremath{\subset}$ implying a quantum subsystem. We find the reduced state in
R2 using two independent ways: (a) Evaluation of the reduced state of
R2 from vacuum state in
M, which yields a thermal density matrix. (b) First, evaluating the reduced state in
R1 from
M yielding a thermal state in
R1, and subsequently evaluating the reduced state in
R2 from
R1. In this article, we attempt to address the question of whether both these independent paths yield the same reduced state in
R2. To that end, we devise a method that involves cleaving the Rindler wedge
R1 into two domains such that they form a thermofield double. One of the domains aligns itself along the wedge
R2 while the other is a diamond-shaped construction between the boundaries of
R1 and
R2. We conclude that both these independent methods yield two different answers, indicating that there are systems in which the reduced quantum state of a subsystem can be path dependent and hence not unique in a quantum field theoretic setting. We discuss the possible implications of our result in the context of quantum states outside a nonextremal black hole formed by collapsing matter.
Abstract We study the entanglement dynamics of thermofield double (TFD) states in integrable spin chains and quantum field theories. We show that, for a natural choice of the Hamiltonian eigenbasis, the TFD evolution may be interpreted as a quantum quench from an initial state which is low-entangled in the real-space representation and displays a simple quasiparticle structure. Based on a semiclassical picture analogous to the one developed for standard quantum quenches, we conjecture a formula for the entanglement dynamics, which is valid for both discrete and continuous integrable field theories, and expected to be exact in the scaling limit of large space and time scales. We test our conjecture in two prototypical examples of integrable spin chains, where numerical tests are possible. First, in the XY-model, we compare our predictions with exact results obtained by mapping the system to free fermions, finding excellent agreement. Second, we test our conjecture in the interacting XXZ Heisenberg model, against numerical iTEBD calculations. For the latter, we generally find good agreement, although, for some range of the system parameters and within the accessible simulation times, some small discrepancies are visible, which we attribute to finite-time effects.
Based on Maldacena's description of an eternal anti--de Sitter (AdS) black hole, we reassess the thermo field dynamics (TFD) formalism in the context of the AdS/CFT correspondence. The model studied here involves the maximally extended AdS-Schwarschild solution and two (noninteracting) copies of the conformal field theory (CFT) associated to the global AdS spacetime, along with an extension of the string by imposing natural gluing conditions in the horizon. We show that the gluing conditions in the horizon define a string boundary state which is identified with the TFD thermal vacuum, globally defined in the Kruskal extension of the AdS black hole. We emphasize the connection of this picture with unitary
SU(1,1) TFD formulation, and we show that information about the bulk and the conformal boundary is present in the
SU(1,1) parameters. Using the unitary
SU(1,1) TFD formulation, a canonical prescription for calculating the world sheet real time thermal Green's function is made, and the entropy associated with the entanglement of the two CFT's is calculated.