PT symmetric field theory

Physical systems are described by a quantity called the Hamiltonian. In conventional
quantum physics two kinds of Hamiltonians are used,
(i) Hermitian Hamiltonians, which govern the behaviour of isolated
systems, and (ii) non-Hermitian Hamiltonians, which have been used to describe
the behaviour of systems in contact with the environment.

Hermitian Hamiltonians describe idealised systems in equilibrium whose total
energy and probability are conserved; the energy levels of such systems are real.

Non-Hermitian Hamiltonians in general receive energy from and/or dissipate energy
into their environment, so they are not typically in equilibrium, their energy and
probability are not conserved, and their energy levels are complex, due to the levels being unstable.

This proposal concerns a category of so-called PT-symmetric
Hamiltonians, which share properties of both Hermitian and non-Hermitian
Hamiltonians, being intermediate between conservative and dissipative systems.
Like non-Hermitian systems,PT-symmetric systems are not isolated, but their contact
with the environment is constrained so that gain from the environment
and loss to the environment are exactly balanced. Thus, while they
are not isolated, PT-symmetric systems in equilibrium behave
like Hermitian systems and their energy levels are real. However,
unlike Hermitian systems, PT-symmetric systems can exhibit
a transition from an unbroken equilibrium phase, where the energies
are real, to a broken nonequilibrium phase where the energies are
complex. Hermitian systems can never have complex energies and thus
cannot have such a phase transition. The PT phase transition is a characteristic
signature that has been observed in experiments.

Quantum mechanics is essential for describing the physics of particles and involves a
finite number of degrees of freedom. However, particles are excitations of quantum fields,
which are defined over all space and time. Quantum field theories
have infinitely many degrees of freedom. Consequently the formulation of PT-symmetric
field theory is required to describe any fundamental theory involving PT symmetry.
Moreover, even within the framework of Hermitian quantum field theories, non-Hermitian
PT symmetric features often emerge in calculations. These features tend to be dismissed,
either on the basis of nonrigorous mathematics related to prescriptions introduced to extract
finite numbers from divergent expressions in calculations,
or incompleteness of the physical model. This proposal investigates directly the role and
properties of PT symmetry in fundamental quantum field theories by investigating the following
questions:

1. Are there analogues in quantum field theory of the features that distinguish PT-symmetric
quantum mechanical systems from Hermitian quantum mechanics?

2.Are there any restrictions on the type of PT-symmetric field theories that show analogous features?

3. Can non-Hermitian features, which arise due to divergences in Hermitian field theories, be dealt
with by procedures within the framework of PT-symmetric quantum field theory?

4. Can PT-symmetric field theories lead to new possibilities for models of fundamental physics, which,
in low number of spatial dimension, may be realised in the laboratory?

These are the questions that the project aims to answer.

The work in this proposal will provide the foundations for a significant extension of the scope of PT symmetry.

PT symmetry is currently a hot area of research in applied science. The published research on PT symmetry
has demonstrated diverse applications in optical systems, lasers, graphene, topological insulators,
superconducting wires, NMR, fluid dynamics, metamaterials, optomechanical systems and
wireless power transfer (Nature 546 (2017) 387).

The proposed study of PT-symmetric quantum field theory will extend significantly the type of systems
which show PT-symmetric behaviour. In particular, starting from a bare Hermitian Lagrangian, the
determination of the conditions in which a PT-symmetric renormalised Lagrangian can arise is a
problem whose solution will have wide ramifications for fundamental physics.

The proposed study will also deepen the understanding of eigenvalue problems in the context of functions of real variables,
whose formulation and solution are vital to much of mathematics- see, for example, J. Phys. A. Math. Theor. 52 (2019) 315202.
For PT-symmetric quantum field theory the analogous problem can be posed in the context of functionals, and is not well understood.
Because of the generality of this problem and its relevance to non-Hermitian physics, the solution of the functional eigenvalue
problem will have a wide applicability in fundamental physics and applied mathematics.

The training that the post-doctoral worker will receive through the research work will enhance
his/her employability because of the highly interdisciplinary nature of the proposal.