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6.1 Solvable systems and perturbation theory

Quantum dynamics is given through a Hamiltonian operator ˆH, which determines the Schrödinger flow ˙ ˆ iℏψ = H ψ, or alternatively Heisenberg equations of motion ˙ˆ −1 ˆ ˆ O = − iℏ [O, H ] for time-dependent operators. Independent of the representation, expectation values obey the equations of motion
d ⟨[ ˆO,Hˆ]⟩ --⟨Oˆ⟩ = -------- . (72 ) dt iℏ
For basic operators, such as ˆq and ˆp in quantum mechanics, we have a system
d ⟨[qˆ,Hˆ]⟩ d ⟨[ˆp, ˆH ]⟩ --⟨ˆq⟩ = -------- , --⟨ˆp⟩ = -------- (73 ) dt iℏ dt iℏ
of coupled equations, which, as in the Ehrenfest theorem, can be compared with the classical equations for q and p. However, the equations (73View Equation) in general do not form a closed set unless the commutators [ˆq,Hˆ] and [pˆ, ˆH ] are linear in the basic operators ˆq and ˆp, such that only expectation values appear on the right-hand side of Equations (73View Equation). Otherwise, one has terms of the form n m ⟨ˆq ˆp ⟩ with n + m > 1, which, in quantum mechanics, are independent of the expectation values ⟨ˆq⟩ and ⟨ˆp⟩. There is no closed set of equations unless one includes the dynamics of all the moments of a state related to the additional terms n m ⟨ˆq ˆp ⟩. A more practical formulation refers to the infinite-dimensional quantum phase space parameterized by expectation values together with the quantum variables Ga,n of Equation (56View Equation). Poisson relations between all these variables follow from expectation values of commutators, and are available in general form [105Jump To The Next Citation Point282Jump To The Next Citation Point]. The flow on this quantum phase space is then simply given by the quantum Hamiltonian defined as the expectation value ˆ HQ = ⟨H ⟩ of the Hamiltonian, interpreted as a function of the quantum variables, which parameterize the state used in the expectation value. Also here, in general, coupling terms between expectation values and quantum variables arise in HQ, which then couple the Hamiltonian equations of motion obtained from H Q for the expectation values and all the quantum variables. But these would be infinitely many coupled ordinary differential equations, which usually is much more complicated to solve than the single partial Schrödinger equation.

Physically, this means that quantum mechanics is described by a wave function, where all the moments, such as fluctuations and deformations from a Gaussian, play a role in the dynamical evolution of expectation values. These quantum variables evolve, and in general couple the expectation values due to quantum backreaction. This quantum backreaction is the reason for quantum corrections to classical dynamics, which is often usefully summarized in effective equations. Such equations, which have the classical form but are amended by quantum corrections, are much easier to solve and more intuitive to understand than the full quantum equations. But they have to be derived from the quantum system in the first place, which does require an analysis of the infinitely-many coupled equations for all moments.

Fortunately, this can often be done using approximation schemes, such as semiclassical expansions and perturbation theory. For this, one needs a treatable starting point as the zeroth order of the perturbation theory, a solvable model, whose equations of motion for moments are automatically decoupled from those for expectation values. Such models are the linear ones, where commutators [⋅,Hˆ] between the basic operators and the Hamiltonian are linear in the basic operators. For a canonical algebra of basic operators such as [ˆq, ˆp] = iℏ, the Hamiltonian must be quadratic, e.g., be that of the harmonic oscillator. Then, the quantum equations of motion for qˆ and ˆp decouple from the moments and form a finite set, which can be easily solved. The harmonic oscillator is not the only quadratic Hamiltonian, but it is special because it allows (coherent state) solutions, whose higher moments are all constant. (This corresponds to the horizontal sections of the quantum phase space mentioned in Section 5.6.2.)

This explains the special role of the harmonic oscillator (or massive free field theories) in perturbation theory: any linear system can be used as zeroth order in a semiclassical perturbation expansion for models, which add interactions, i.e., non-quadratic terms, to the Hamiltonian and thus coupling terms between expectation values and quantum variables. In general, this results in higher-dimensional effective systems, where some of the quantum variables, but only finite ones, may remain as independent variables. If one perturbs around the harmonic oscillator, on the other hand, the fact that it allows solutions (especially the ground state) with constant quantum variables, allows an additional approximation, using adiabatic behavior of the quantum variables. In this case, one can even solve for the remaining quantum variables in terms of the classical variables, which provides explicit effective potentials and other corrections without having to refer to quantum degrees of freedom. In this way, for instance, one can derive the low-energy effective action well known from particle physics [105Jump To The Next Citation Point282]. This behavior, however, is not general because not all linear systems allow the adiabaticity assumption. For instance, a free particle, whose Hamiltonian is also quadratic, violates the adiabaticity assumption. Correspondingly, quantum terms in the low-energy effective action diverge when the frequency of the harmonic oscillator approaches zero, a fact that is known as infrared divergence in quantum field theory. Higher dimensional effective systems, where some quantum variables remain as independent degrees of freedom, still exist. Similar techniques are applied to cosmological structure formation in [104], with possible applications to non-Gaussianities.

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