# Functional integrals and inequivalent representations in Quantum Field Theory

## Abstract

We discuss canonical transformations in Quantum Field Theory in the framework of the functional-integral approach. In contrast with ordinary Quantum Mechanics, canonical transformations in Quantum Field Theory are mathematically more subtle due to the existence of unitarily inequivalent representations of canonical commutation relations. When one works with functional integrals, it is not immediately clear how this algebraic feature manifests itself in the formalism. Here we attack this issue by considering the canonical transformations in the context of coherent-state functional integrals. Specifically, in the case of linear canonical transformations, we derive the general functional-integral representations for both transition amplitude and partition function phrased in terms of new canonical variables. By means of this, we show how in the infinite-volume limit the canonical transformations induce a transition from one representation of canonical commutation relations to another one and under what conditions the representations are unitarily inequivalent. We also consider the partition function and derive the energy gap between statistical systems described in two different representations which, among others, allows to establish a connection with continuous phase transitions. We illustrate the inner workings of the outlined mechanism by discussing two prototypical systems: the van Hove model and the Bogoliubov model of weakly interactingmore »

- Authors:

- Dipartimento di Fisica, Università di Salerno, Via Giovanni Paolo II, 132 84084 Fisciano (Italy)
- (Italy)
- FNSPE, Czech Technical University in Prague, Břehová 7, 115 19 Praha 1 (Czech Republic)

- Publication Date:

- OSTI Identifier:
- 22701515

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics

- Additional Journal Information:
- Journal Volume: 383; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANNIHILATION OPERATORS; BOSE-EINSTEIN GAS; CANONICAL TRANSFORMATIONS; COMMUTATION RELATIONS; ENERGY GAP; INTEGRALS; PARTITION FUNCTIONS; PHASE TRANSFORMATIONS; QUANTUM FIELD THEORY; QUANTUM MECHANICS

### Citation Formats

```
Blasone, M., E-mail: blasone@sa.infn.it, INFN Sezione di Napoli, Gruppo collegato di Salerno, Jizba, P., E-mail: p.jizba@fjfi.cvut.cz, Smaldone, L., E-mail: lsmaldone@sa.infn.it, and INFN Sezione di Napoli, Gruppo collegato di Salerno.
```*Functional integrals and inequivalent representations in Quantum Field Theory*. United States: N. p., 2017.
Web. doi:10.1016/J.AOP.2017.05.022.

```
Blasone, M., E-mail: blasone@sa.infn.it, INFN Sezione di Napoli, Gruppo collegato di Salerno, Jizba, P., E-mail: p.jizba@fjfi.cvut.cz, Smaldone, L., E-mail: lsmaldone@sa.infn.it, & INFN Sezione di Napoli, Gruppo collegato di Salerno.
```*Functional integrals and inequivalent representations in Quantum Field Theory*. United States. doi:10.1016/J.AOP.2017.05.022.

```
Blasone, M., E-mail: blasone@sa.infn.it, INFN Sezione di Napoli, Gruppo collegato di Salerno, Jizba, P., E-mail: p.jizba@fjfi.cvut.cz, Smaldone, L., E-mail: lsmaldone@sa.infn.it, and INFN Sezione di Napoli, Gruppo collegato di Salerno. Tue .
"Functional integrals and inequivalent representations in Quantum Field Theory". United States. doi:10.1016/J.AOP.2017.05.022.
```

```
@article{osti_22701515,
```

title = {Functional integrals and inequivalent representations in Quantum Field Theory},

author = {Blasone, M., E-mail: blasone@sa.infn.it and INFN Sezione di Napoli, Gruppo collegato di Salerno and Jizba, P., E-mail: p.jizba@fjfi.cvut.cz and Smaldone, L., E-mail: lsmaldone@sa.infn.it and INFN Sezione di Napoli, Gruppo collegato di Salerno},

abstractNote = {We discuss canonical transformations in Quantum Field Theory in the framework of the functional-integral approach. In contrast with ordinary Quantum Mechanics, canonical transformations in Quantum Field Theory are mathematically more subtle due to the existence of unitarily inequivalent representations of canonical commutation relations. When one works with functional integrals, it is not immediately clear how this algebraic feature manifests itself in the formalism. Here we attack this issue by considering the canonical transformations in the context of coherent-state functional integrals. Specifically, in the case of linear canonical transformations, we derive the general functional-integral representations for both transition amplitude and partition function phrased in terms of new canonical variables. By means of this, we show how in the infinite-volume limit the canonical transformations induce a transition from one representation of canonical commutation relations to another one and under what conditions the representations are unitarily inequivalent. We also consider the partition function and derive the energy gap between statistical systems described in two different representations which, among others, allows to establish a connection with continuous phase transitions. We illustrate the inner workings of the outlined mechanism by discussing two prototypical systems: the van Hove model and the Bogoliubov model of weakly interacting Bose gas. - Highlights: • Functional integrals are sensitive to representations of the Heisenberg–Weyl algebra. • Inequivalent representations can be exhibited in the functional integral formalism. • Passage among inequivalent representations is seen as second-order phase transition. • The inequivalence of representations is physically marked by an infinite energy gap.},

doi = {10.1016/J.AOP.2017.05.022},

journal = {Annals of Physics},

issn = {0003-4916},

number = ,

volume = 383,

place = {United States},

year = {2017},

month = {8}

}