Contents

Contents

Idea

In perturbative quantum field theory, the term adiabatic switching refers to considering a smooth transition between vanishing and non-vanishing interaction coupling: the interactions is slowly, hence (borrowing a term from thermodynamics) “adiabatically”, switched on or off. This is mostly a mathematical device, not meant to directly reflect a physical situation of changing coupling, but it does serve to construct physical quantities. This is closely related to the role of operator-valued distributions which are quantities that give well defined linear operators (hence quantum observables) only when evaluated on any bump function.

Originally adiabatic switching was considered (Lippmann-Schwinger 50) only in the time-direction (for a fixed choice of time on Minkowski spacetime) by multiplying the interaction term of the Lagrangian density/Hamiltonian by the exponential $\exp(- \epsilon {\Vert t \Vert})$ (for $\epsilon \in (0,\infty)$ a positive real number and for ${\Vert t\Vert}$ the absolute value of the time coordinate). Review is for instance in (Strocchi 13, section 6.3).

Using this, the Gell-Mann and Low formula (Gell-Mann & Low 51, see Molinari 06) expresses the eigenstates $\vert \psi \rangle$ of an interacting Hamiltonian $H = H_{free} + H_{int}$ in terms of the eigenstates $\vert \Psi_{free} \rangle$ of the free Hamiltonian by the “adiabatic limit”

$\vert \Psi^{\pm}_{int} \rangle \;\propto\; \underset{\epsilon \to 0}{\lim} S_\epsilon(0, \pm \infty) \vert \Psi_{free} \rangle$

(if the limit exists) where $S_\epsilon$ denotes the S-matrix of the adiabatically switched Hamiltonian $H_\epsilon \coloneqq H_{free} + e^{- \epsilon {\Vert t\Vert}}H_{int}$.

More generally, one may consider adiabatic switching taking place not just in time, but in all of spacetime. This the basis of causal perturbation theory and locally covariant perturbative quantum field theory:

In the construction of perturbative quantum field theory via the method of causal perturbation theory the interaction terms $L_{int}$ used in the mathematical construction of the S-matrix are multiplied with a “coupling constant$g$ which is in fact taken to be a smooth function of compact support on spacetime, hence a bump function:

$L_g = L_{free} + g L_{int} \,.$

This means that the the interaction as modeled by the S-matrix

$S_g \coloneqq T \exp( \tfrac{i}{\hbar} \int_{X} g :L_{int}(x): )$

is non-trivial only on a compact subspace of spacetime, towards its boundary it smoothly drops to zero. Hence outside this region the interaction is “switched off”.

Since the actual interactions in physics are of course not “switched off” anywhere, the use of an adiabatic switching is just an intermediate mathematical step. Originally in (Epstein-Glaser 73) the idea was that after having constructed the S-matrix for any adiabatic switching $g$, the limit (“adiabatic limit”) $g \to 1$ had to be taken to remove the switching in the end. Failure of this limit to exist is interpreted as “infrared divergency” of the perturbative quantum field theory (since the divergency comes from large scales, hence long wavelength).

But as observed in (Il’in-Slavnov 78) and rediscovered in (Brunetti-Fredenhagen 00), an adiabatic switching map that is unity on a globally hyperbolic sub-spacetime $O \subset X$ is sufficient to compute the perturbative interacting field algebra, hence the algebra of quantum observables $A(O)$ on that subspace, and the collection of all of these as $O$ ranges forms a causally local net of observables which fully captures the quantum field theory in the sense of the Haag-Kastler axioms (this prop.). This perspective is now known as locally covariant algebraic quantum field theory.

The limit of the perturbative S-matrix as the adiabatic switching is removed (if it exists) is called the adiabatic limit or strong adiabatic limit.

If one just asks that the corresponding limit exists for the n-point functions one speaks of a weak adiabatic limit.

Even with the adiabatically switched S-matrix elements (not taking a limit) the local net of quantum observables is well defined (this prop.), this is hence a functor

$\mathcal{O} \mapsto \mathcal{A}(\mathcal{O})$

that assigns algebras of observables to causally closed subsets of spacetime. The colimit algebra

$\mathcal{A} \coloneqq \underset{\underset{\mathcal{O}}{\longrightarrow}}{\lim} \mathcal{A}(\mathcal{O})$

over this functor (in the sense of category theory) always exists. This is also called the algebraic adiabatic limit.

(See around Duch 17, section 4 for review of strong, weak and algebraic adiabaitc limit; and Duch 17, chapter II for results on the weak adiabatic limit)

Here

1. the algebraic adiabatic limit defines the quantum observables in the limit;

2. the weak adiabatic limit may serve to define also the states, hence the interacting vacuum (Duch 17, p. 113-114).

References

The concept of adiabatic switching in the time direction was introduced in

reviewed for instance in

• Franco Strocchi, section 6.3 of An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press, 2013

and the corresponding formula for the interacting eigenstates in terms of the free ones is due to

see

The generalization to switching in all space-time directions was considered for the construction of causal perturbation theory in

The observation that this in fact makes causal perturbation theory a tool for constructing local nets of observables for locally covariant perturbative quantum field theory is due to

• V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32.

• Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661,2000 (math-ph/9903028)

The term “algebraic adiabatic limit” for the resulting local net of observables (or its inductive limit) appears in

• Klaus Fredenhagen, Falk Lindner, p. 7 of Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014 (arXiv:1306.6519)

The weak adiabatic limit in causal perturbation theory for massive fields was shown to exists in

Extension of this result to quantum electrodynamics and phi^4 theory was given in

• P. Blanchard and R. Seneor, Green’s functions for theories with massless particles (in perturbation theory), Ann. Inst. H. Poincaré Sec. A 23 (2), 147–209 (1975) (Numdam)