Bogoliubov transformations and particle definitions

For a free real Klein–Gordon field on a suitable globally hyperbolic background, the conserved Klein–Gordon product is evaluated on a Cauchy surface with future-directed surface element. With signature (−,+,+,+), choose positive-norm modes u and negative-norm conjugates u star. A complete decomposition of the solution space into these two sectors permits an expansion of the field in annihilation and creation operators. The choice of positive-frequency subspace defines the associated vacuum.

A second normalized basis v can contain both u and its conjugate. Its coefficients alpha and beta are restricted by the Klein–Gordon products so that canonical commutators are preserved. For a finite collection of modes these are symplectic normalization conditions. In an infinite field theory, convergence and unitary implementability add further requirements; an unrestricted formal infinite transformation need not act as a unitary operator on one common Fock space.

Taking the Klein–Gordon product of v with the field gives an annihilation operator containing the old annihilation operator and a creation part with a minus beta conjugate. The old vacuum is therefore not generally annihilated by the new operator. Its mean new particle number is the squared norm of beta. This result follows from the old vacuum commutators and does not require a classical trajectory for an emitted particle.

For one bosonic mode, the normalization says that alpha squared in magnitude exceeds beta squared by one. A squeezing parametrization automatically satisfies it, but a generic squeezed pure state is not a thermal density matrix. Thermal behavior needs additional structure, such as a particular relation across frequencies and a restriction to an accessible subsystem. A nonzero mean number alone does not specify number fluctuations, correlations, entropy, or a temperature.

Two inertial observers related by an ordinary Lorentz transformation in Minkowski spacetime agree on its vacuum; a coordinate relabeling alone does not create particles. The interesting comparisons involve distinct positive-frequency structures, for example early and late stationary regions separated by a changing geometry, or a boost-generated time evolution in an accelerated wedge. An operational detector additionally depends on its trajectory, coupling, switching, and the state's correlation function.

For a black-hole calculation, the early incoming state, late outgoing modes, and treatment of modes crossing the horizon must be stated. Regularity at a horizon and absence of incoming flux select different physical properties from a static vacuum condition at infinity. Renormalized stress tensors can be defined without assigning a universal local particle number, and their value is not determined by beta for one chosen mode alone. Maintaining these distinctions prevents a basis calculation from becoming an unsupported claim that every observer sees the same thermal bath.