We analyze a quantum central limit theorem without centering and a quantum Poisson limit theorem on an array of indistinguishable quanta and investigate the emergence of the continuous and discrete structure of the quantized electromagnetic field in classical states. Moreover, we identify the observables that converge to the random variables which induce the mixing measure of a classical state. Finally, we show that the transformation to normal order is a conditional expectation replacing expectations of functions of the annihilation and creation operator by expectations with respect to global classical observables of the array.
We analyze a quantum central limit theorem without centering and a quantum Poisson limit theorem on an array of indistinguishable quanta and investigate the emergence of the continuous and discrete structure of the quantized electromagnetic field in classical states. Moreover, we identify the observables that converge to the random variables which induce the mixing measure of a classical state. Finally, we show that the transformation to normal order is a conditional expectation replacing expectations of functions of the annihilation and creation operator by expectations with respect to global classical observables of the array.
At the end of the preceding chapter, we showed that the photon number states | n 〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈 n | Ê x | n 〉 = 0 no matter how large the value of n . We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see. Eigenstates of the annihilation operator and minimum uncertainty states In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.
Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.
Abstract Classical Koopman–von Neumann Hilbert spaces of states are constructed here by the action of classical random fields on a vacuum state in ways that support an action of the quantized electromagnetic field and of the U (1)—invariant observables of the quantized Dirac spinor field, allowing a manifestly Lorentz invariant classical understanding of the state spaces of the two field theories, generalizing the Quantum–Mechanics–Free Systems of Tsang&Caves and Quantum Non-Demolition measurements. The algebra of functions on a classical phase space is commutative but the algebra of classical observables associated with coordinate transformations is noncommutative, so that, for example, we can as much ask whether a classical state is an eigenstate of a rotation as we can in quantum mechanics and so that entangled states can be distinguished from mixed states, making classical random fields as weird as quantum fields.
Classical dynamics and the classical concept of space-time reality is based on the assumption that the objects of physics can be observed continuously. The discrete structure of matter (atoms) and fields (quanta), however, implies that the process of observation is quantized. In this paper we discuss the consequences of this paradigm change from continuity to discreteness and from determinism to chance. By taking into account the quantized structure of observation, we are led to the conclusion that the classical concept of reality has to be replaced by a quantum concept of observability. It implies that quantum dynamics is not a generalization of classical dynamics. Rather, the two theories apply to complementary idealizations of nature representing opposite extremes on a scale of observability. They are related by correspondence rules. Both theories disregard experimental noise. Statistical and thermal noise provide the experimental foundation for statistical physics.
We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained hermitean operators consisting of non-commuting variables. Thanks to the hermicity constraints, we obtain positive-definite dis- tribution for the expectation values of observables. These probability distributions open some pathway for an emergence of classical behaviours in the limit of innitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probabil- ity distributions so derived also enable us to evaluate the nontrivial time-dependence of certain differential entropies.
Semiclassical models can be used, see e.g. Scully and Zubairy [Quantum Optics; Cambridge University Press: Cambridge, 1997], to simulate some distinguishing features of quantum optics. In this paper we show that one can also proceed successfully in the opposite direction. A quantum-like model for the classical random electromagnetic field is proposed. Averages with respect to classical random fields can be approximated by quantum-like averages given by the von Neumann trace-formula: . The operators ρ and can be easily determined as classical quantities. Here ρ is obtained by normalization of the covariance operator of the random field. It has all the features of the von Neumann density operator. Additionally an analogue of the quantum observable is given by the second derivative of a functional (‘classical variable’) f (E, B) of the classical electromagnetic field ψ = (E, B).