Nothing's plenty

With a tundric wind driving intermittent squalls of rain across the barren agriscape outside, I decided to stay inside today and re-read Ian Aitchison's excellent 1985 review of the concept of the vacuum in quantum field theory, (Contemporary Physics , Vol.26, No.4, pp333-391).

In the popular literature, the vacuum of quantum field theory is characterised as a seething torrent of evanescent 'virtual' particles, popping in and out of existence. It is crucial to appreciate, however, that there are different types of vacuum within quantum field theory. Firstly, there is the vacuum of a free field. The states of a free field are represented in quantum field theory by the elements of a vector space called a Fock space. The vacuum state of a free field is represented by a vector in Fock space with unique properties, called the vacuum vector. Then there is the vacuum state of an interacting field, which is not represented by the vacuum vector in any Fock space. In quantum electrodynamics, for example, there is: (i) the vacuum state of the electromagnetic field; (ii) the vacuum state of the electron field; and (iii) the vacuum state of the interacting electron-electromagnetic field, and all three are distinct.

The popular notion of the vacuum, with its innumerable collection of virtual particles, corresponds to the interacting vacuum, sometimes also called the 'dressed' vacuum. The experimentally detectable effects ascribed to the dressed vacuum are referred to as 'vacuum fluctuations'. However, the free-field vacuum is the only vacuum which is theoretically well-defined in quantum field theory. As Prugovecki states, "the actual computations performed in perturbation theory actually begin with expressions for [free-field states] states,...formulated in Fock space, and then progress through a chain of computations dictated by Feynman rules, which have no direct bearing to a mathematically rigorous realization of a non-Fock representation...Hence, in conventional QFT [quantum field theory] the existence of...a corresponding unique and global 'dressed vacuum', is merely a conjecture rather than a mathematical fact," (Principles of Quantum General Relativity, World Scientific, 1995, p198-199). Rugh and Zinkernagel concur, arguing that the popular picture of the production and annihilation of virtual particles in the 'interacting' vacuum, "is actually misleading as no production or annihilation takes place in the vacuum. The point is rather that, in the ground state of the full interacting field system, the number of quanta (particles) for any of the fields is not well-defined," (The quantum vacuum and the cosmological constant problem, 2002, p12, footnote 27.)

Aitchison's approach to virtual particles is to argue that they are basically a consequence of the so-called perturbative approach to the representation of interactions in quantum field theory. In quantum theory, the total energy of a system is represented by something called the Hamiltonian H. This Hamiltonian can be broken into a free Hamiltonian H₀ and an interaction Hamiltonian H_I:

H = H₀ + H_I

In quantum field theory, the free Hamiltonian is an operator defined upon Fock space. In quantum theory generally, an eigenstate of some operator represents a state of the system in which the quantity represented by that operator possesses a definite value. The vacuum vector in Fock space is the lowest energy eigenstate of the free Hamiltonian. There is another operator on Fock space called the particle number operator, and the vacuum vector is the zero eigenstate of this operator, indicating the presence of zero number of particles. Hence, the vacuum vector in Fock space represents both the lowest energy state of the free field, and the state in which zero particles exist. It is also the unique vector which is invariant under space-time translations; this guarantees that the choice of vacuum state is not dependent upon the position in space or time which the observer occupies.

Now, in perturbative quantum field theory, interactions are represented in Fock space, the state space for the free field, by also defining the interaction Hamiltonian H_I as an operator on the Fock space. Aitchison argues that a (perturbative) interacting vacuum is an eigenstate of the full Hamiltonian H defined on the free field Fock space, and is therefore a superposition of the eigenstates of the free field Hamiltonian H₀. The various eigenstates of the free field Hamiltonian are also eigenstates of the particle number operator, but an eigenstate of the full Hamiltonian is not. The number of particles in the ground state of the interacting system does not itself possess a definite number of particles, but rather is a superposition of all the definite particle number eigenstates of the free field Hamiltonian. According to Aitchison, the interacting vacuum contains arbitrary numbers of virtual particles only in the sense that the interacting vacuum is a superposition of definite particle number eigenstates of the free Hamiltonian. The so-called fluctuations of the interacting vacuum, are fluctuations over the free field eigenstates within the superposition. Virtual particle transitions appear not to conserve energy-momentum, and never appear in the incoming and outgoing states of a reaction, because reactions in perturbative quantum field theory are represented as transitions from free field eigenstates to interacting field eigenstates, and back again.

This is extremely illuminating, but Aitchison's interpretation needs to be treated with some caution. For a start, Haag's theorem demonstrates that a free-field Fock space cannot directly represent the interacting field vacuum, (see Earman and Fraser for an excellent discussion). Haag's theorem basically proves that a Fock space cannot possess the vacuum vector of a free-field and the vacuum vector of an interacting field. A vacuum vector is required to be invariant under space-time translations, and a Fock space possesses, up to phase, a unique translation-invariant vector. The Fock space vacuum vector cannot be the ground state of both the free Hamiltonian and the full Hamiltonian, and the vacuum vector is the unique vector invariant under space-time translations.

The ongoing absence of a mathematically well-defined representation of interacting quantum field systems, and the failure of string theory to provide a viable alternative, makes it very difficult to say anything meaningful about what the interacting quantum vacuum really is.

Vacuum energy Fock space Virtual particles