Minkowski Vacuum State Research Articles

Energy density-flux correlations in an unusual quantum state and in the vacuum

In this paper we consider the question of the degree to which negative and positive energy are intertwined. We examine in more detail a previously studied quantum state of the massless minimally coupled scalar field, which we call a ''Helfer state.'' This is a state in which the energy density can be made arbitrarily negative over an arbitrarily large region of space, but only at one instant in time. In the Helfer state, the negative energy density is accompanied by rapidly time-varying energy fluxes. It is the latter feature which allows the quantum inequalities, bounds which restrict the magnitude and duration of negative energy, to hold for this class of states. An observer who initially passes through the negative energy region will quickly encounter fluxes of positive energy which subsequently enter the region. We examine in detail the correlation between the energy density and flux in the Helfer state in terms of their expectation values. We then study the correlation function between energy density and flux in the Minkowski vacuum state, for a massless minimally coupled scalar field in both two and four dimensions. In this latter analysis we examine correlation functions rather than expectation values. Remarkably, we see qualitativelymore » similar behavior to that in the Helfer state. More specifically, an initial negative energy vacuum fluctuation in some region of space is correlated with a subsequent flux fluctuation of positive energy into the region. We speculate that the mechanism which ensures that the quantum inequalities hold in the Helfer state, as well as in other quantum states associated with negative energy, is, at least in some sense, already encoded in the fluctuations of the vacuum.« less

The Unruh effect revisited

We give a complete and rigorous proof of the Unruh effect, in the following form. We show that the state of a two-level system, uniformly accelerated with proper acceleration a and initially coupled to a scalar Bose field initially in the Minkowski vacuum state will converge, asymptotically in the detector's proper time, to the Gibbs state at inverse temperature . The result also holds if the field and detector are initially in an excited state. We treat the problem as one of return to equilibrium, exploiting in particular that the Minkowski vacuum is a KMS state with respect to Lorentz boosts. We then use the recently developed spectral techniques to prove the stated result.

Supersymmetry and stability of flux vacua

We describe a modified KKLT mechanism of moduli stabilization in a supersymmetric Minkowski vacuum state. In this mechanism, supersymmetry ensures vacuum stability and positivity of the mass matrix for the dilaton, complex structure, and the volume modulus.

ON THE ENTANGLED STATE REPRESENTATION, EXCITATION REPRESENTATION AND EFFECTIVE NUMBER-PHASE STATE REPRESENTATIONS IN RINDLER SPACE

We analyze the Unruh effect from the point of view of quantum entanglement. We introduce the entangled state representation in Rindler space and show that the Minkowski vacuum state is an entangled state in Rindler space. The corresponding squeezing operator, which is related to the acceleration of the detector, is obtained naturally. The excitation representation and number state–phase state representations are also introduced in Rindler space. The number-phase commutative relation is established.

Fulling-Unruh effect in general stationary accelerated frames

We study the generalized Unruh effect for accelerated reference frames that include rotation in addition to acceleration. We focus particularly on the case where the motion is planar, with the presence of a static limit in addition to the event horizon. Possible definitions of an accelerated vacuum state are examined and the interpretation of the Minkowski vacuum state as a thermodynamic state is discussed. Such a thermodynamic state is shown to depend on two parameters, the acceleration temperature and a drift velocity, which are determined by the acceleration and angular velocity of the accelerated frame. We relate the properties of the Minkowski vacuum in the accelerated frame to the excitation spectrum of a detector that is stationary in this frame. The detector can be excited both by absorbing positive energy quanta in the ``hot'' vacuum state and by emitting negative energy quanta into the ``ergosphere'' between the horizon and the static limit. The effects are related to similar effects in the gravitational field of a rotating black hole.

Generalized Uncertainty Relation of One-Dimensional Rindler Oscillator

General Minkowski vacuum state is seen to be equivalent to a thermal bath for a Rindler uniformly accelerated observer. This paper calculates the generalized uncertainty relation of one-dimensional Rindler oscillator in the coordinate representation. The calculations show that for a Rindler uniformly accelerated observer there is not only general quantum fluctuation but also thermal fluctuation related to his acceleration.

The rotatingdetector and vacuum fluctuations

In this work we compare the quantization of a massless scalar field inan inertial frame with the quantization in a rotating frame. We usedthe Trocheries-Takeno mapping to relate measurements in the inertialandtherotating frames. An exact solution of the Klein-Gordon equation in therotating coordinate system is found and the Bogolubov transformationbetween the inertial and rotating modes is calculated, showing that therotating observer defines a vacuum state different from the Minkowskione.We also obtain the response function of an Unruh-De Witt detectorcoupledwiththe scalar field travelling in a uniformly rotating worldline. The responsefunction is obtained for two different situations: when the quantum fieldisprepared in the usual Minkowski vacuum state and when it is prepared intheTrocheries-Takeno vacuum state. We also consider the case of an inertialdetector interacting with the field in the rotating vacuum.

Quantum field and uniformly accelerated oscillator

We present an exact treatment of the influences on a quantum scalar field in its Minkowski vacuum state induced by coupling of the field to a uniformly accelerated harmonic oscillator. We show that there are no radiation from the oscillator in the point of view of a uniformly accelerating observer. On the other hand, there are radiations in the point of view of an inertial observer. It is shown that Einstein-Podolsky-Rosen (EPR) like correlations of Rindler particles in Minkowski vacuum states are modified by a phase factor in front of the momentum-symmetric Rindler operators. The exact quantization of a time-dependent oscillator coupled to a massless scalar field was given.

Influence of uniformly accelerated oscillator on quantum field.

The influence on a quantum scalar field in its Minkowski vacuum state induced by weak coupling of the field to a uniformly accelerated harmonic oscillator is studied. From the perspective of a Rindler observer, the reaction of the oscillator on the quantum field distorts the Planckian particle spectrum of the Minkowski vacuum state and induces a change of state for the quantum field even without simultaneous excitation of the oscillator. From the perspective of an inertial observer, apart from inducing the excitation of a Minkowski vacuum state without excitation of the oscillator, the reaction of the oscillator on the field also gives rise to the presence of particles in the Minkowski vacuum state. In the long time limit, when the oscillator comes into thermal equilibrium again with the Minkowski vacuum state according to a Rindler observer, a closed expression for the particle number is worked out and its physical implications are discussed. EPR correlation of the Minkowski vacuum state for a Rindler observer is also disturbed by the presence of an accelerated oscillator even in the long time limit. When the oscillator is viewed as a simple model of a measuring device, the result suggests that noise induced by measurementmore » on a quantum field might present difficulties for a Rindler observer who wants to uphold EPR correlation across the event horizon. {copyright} {ital 1996 The American Physical Society.}« less

Exact Solutions of Quantum Massive Spin-1 Field Equations and Their Application to the Hawking-Unruh Effect

A generalisation of the Stueckelberg aCtion is implemented to describe spin-1 fields propagating in a four-dimensional Rindler manifold. Exact solutions of the field equations are obtained and the expectation value in the Minkowski vacuum state of the Rindler particle number operator is evaluated in the massive and massless cases. Further on, the energy-momentum tensor expectation value in the Rindler vacuum is computed and its qualitative behavior is analysed. In the zero mass limit we rederive the results obtained by other methods. Like this case, in the massive one it is shown that the energy-momentum spectrum is not Planckian. For small values of the mass, this spectrum is characterised by the Bose distribution function apart from a modification generated by a reduced density of states factor which is explicitly calculated.

Inertial and noninertial particle detectors and vacuum fluctuations.

In this paper the Glauber and Unruh-DeWitt detector models are discussed. The Unruh-DeWitt detector model is adopted. The process of measurement and the probability of excitation for the inertial and noninertial detectors interacting with the Minkowski vacuum state of the field are studied. This probability is evaluated for a finite time interval.

Thermal bath and decoherence of Rindler spacetimes.

The Minkowski vacuum state is a thermal state as far as any uniformly accelerated observer is concerned. However, if we allow a thermal bath, or any system, to come to equilibrium with that thermal state, the state is not left as the Minkowski vacuum. The thermal bath, or the scatterer, destroys crucial coherences in the state across the horizon which destroy the character of the state.

Quantum states of a field partitioned by an accelerated frame.

The discrete Minkowski Bessel mode technology is developed. This development includes (i) the eigenvalue spectrum, (ii) the phase-space integral formulation of the quantized Klein-Gordon (KG) field in accelerated frames with a finite bottom ($b>0$), and (iii) the procedure for making the transition to bottomless ($b=0$) frames. This technology is used to show that the Rindler vacuum state of an acceleration-partitioned real KG field is perpendicular not only to the Minkowski vacuum state, but also to every quantum state associated with a "detector" accelerating through what initially was the Minkowski vacuum. In fact, every element of the Hilbert space containing the Rindler vacuum is perpendicular to the Hilbert space containing the Minkowski vacuum.

Rindler observers, correlated states, boundary conditions, and the meaning of the thermal spectrum.

Two new results concerning the Minkowski vacuum state as seen by Rindler observers are derived. The first result is that the thermal excitations detected by two causally separated Rindler observers are correlated in both energy and phase. Therefore, the state as seen by these observers is similar to an Einstein-Podolsky-Rosen state. The second result suggests that a Rindler observer detects the same thermal spectrum even if there is a boundary condition beyond his horizon. A discussion of what these effects tell us about the origin of the thermal spectrum follows.

Hyperspherical Rindler space, dimensional reduction, and de Sitter-space scalar field theory.

Field theory in de Sitter space is related to flat-spacetime theory via dimensional reduction. de Sitter space can be embedded in Minkowski space of one higher dimension as a hyperboloid of one sheet. This paper examines the field theory imposed on de Sitter space by a free, massless, real scalar field in the embedding space. This corresponds to a collection of massive de Sitter-space scalars, of all masses above a positive lower bound: Half of the field normal modes in hyperspherical Rindler coordinates specialize to such fields on an embedded de Sitter space, while the other half vanish there. Dimensional reduction simply eliminates those noncontributing modes. The quantized field theory in hyperspherical Rindler space, hence, in de Sitter space, is treated here using both canonical operator expansions and the functional Schr\"odinger formalism. Transformations between field operators and state wave functionals in the Rindler--de Sitter theory and those in the Minkowski-space formulation are calculated explicitly. The Minkowski vacuum state corresponds to the Chernikov-Tagirov or Euclidean vacuum of the de Sitter-space fields. The Rindler--de Sitter `` particle'' content of this state, for various simple mode choices, can be time dependent, even nonmonotonic; it can also exhibit ``thermal'' features recalling those of rectangular Rindler-space theory or of Euclidean field theory in de Sitter space.

Horizons and Generating Functionals for the Temperature Green Functions

The generating functional approach to Green functions in the thermal equilibrium is used to explore the geometrical origin of the temperatures of the quantum fields in the Rindler space-time and black hole spacetimes. It is shown that under the transformation from Minkowski space to the Rindler space the path integral representation for the Euclidean generating functionals of Green functions at zero temperature would transform into the corresponding ones of the quantum fields at a certain finite temperature, and the Minkowski vacuum state would have the same properties as that of the quantum mixed state at the same temperature. All thermal Green functions for the mixed state are given. Similar results would be obtained for the Schwarzschild, the Reissner-Nordström and the Kerr black holes and whereupon the Hawking temperature for the black holes would have geometrical origin as well as that in the Rindler spacetime. The various density operators of the mixed states at the Hawking temperature for the black hole spacetimes are specified.

HORIZONS AND GENERATING FUNCTIONALS FOR THE TEMPERATURE GREEN'S FUNCTIONS

The generating functional approach to Green's functions in the thermal equilibrium is applied to explore the geometrical origin of the temperatures of the quantum fields in the Rindler spacetime and black hole spacetimes. It is shown that under the transformation from Minkowski space to the Rindler space the path integral representation for the Euclidean generating functionals of Green's functions at zero temperature would transform into the corresponding ones of the quantum fields at a certain finite temperature, and the Minkowski vacuum state would have the same properties as that of the quantum mixed state at that temperature. All thermal Green's functions for the mixed state are given. Similar results would be obtained for the Schwarzschild, the Reissner-Nordstrom and the Kerr black holes and whereupon the Hawking temperature for the black holes would have geometrical origin as well as that in the Eindler spacetime. The various density operators of the mixed states at the Hawking temperature for the black hole spacetimes are specified.