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K. P. Satheesh
Functional Schrödinger picture was found on the fact that quantum states and
operators can be projected on the field amplitude basis. Historically the quantum
mechanics put forward by Schrödinger is immensely successful in its predictions
about the behaviour of particles at quantum level. But for quantizing classical and
quantum fields we conventionally do not use Schrödinger equation explicitly. In
this review, I treat quantum theory of fields by extending the concepts and methods of Schrödinger approach to quantum mechanics [1]. In quantum mechanics, in
Schrödinger picture, the time dependence of observables is encoded in the states while
the operators remain time independent. In quantum field theory using Schrödinger
picture, this implies a time-independent field operator ˙ ˆ
φ = 0. Canonical quantization
is implemented by demanding commutation relations for conjugate operators. In the
coordinate representation, position operators are represented by their eigenvalues
and momentum operators by differential equations and states by wave functions. In
field theory, we use field operators instead of position operators whose eigenvalues
are functions instead of numbers. The states are represented by wave functional and
Schrödinger equations become functional differential equations.
The techniques which are already tested in quantum mechanics can be used here.
Schrödinger equation describes the time evolution of wave functions which are the
coordinate representations of state vectors. Time-dependent Schrödinger equation
determines their unitary time evolution. Dynamical variables are treated as Hermitian
operators.
In conventional QFT, both fields and their conjugate momenta are treated as operators and used to define the Hamiltonian of the field through a Legendre transformation
of the Lagrangian. In the present treatment, fields are treated as coordinate functional
and Schrödinger equation finds an extension to obtain the functional Schrödinger
equation. Thus, QFT is basically a solution of functional Schrödinger equation.
An exact solution is usually obtained when we treat a free field but approximations are always required when we handle interacting fields. One of the simplest
approximations popularly used in quantum mechanics is the variational approximation where a trial state is expressed in terms of certain parameters which can be later
extremized to obtain bounds of relevant eigenvalues.
In functional Schrödinger picture also I employ an extension of this procedure
which is a Gaussian trial wave functional with variational parameters to implement
variational approximation. Here the study is restricted to scalar fields only but can
be extended to fermionic and gauge fields with slight modifications. Recently, great
interest in the Schrödinger picture formalism of QFT is generated in analysing QFT
in curved space-time. Nice features of the vacuum state (vacuum wave functional)
in this approach avoid several difficulties associated with QFT in curved space-time
[1–4]. The method developed can be comfortably extended to finite temperature QFT
[5]. This aspect is not discussed in this review.
The method discussed can be extended to study the de-coherence of massive fields
during inflation using variance of non-Gaussian exponent in the wave functional. It
is also used to analyse quantum-to-classical transition due to quantum de-coherence
[6, 7]. Schrödinger picture is very well suited to explore properties of the vacuum
K. P. Satheesh
Functional Schrödinger picture was found on the fact that quantum states and
operators can be projected on the field amplitude basis. Historically the quantum
mechanics put forward by Schrödinger is immensely successful in its predictions
about the behaviour of particles at quantum level. But for quantizing classical and
quantum fields we conventionally do not use Schrödinger equation explicitly. In
this review, I treat quantum theory of fields by extending the concepts and methods of Schrödinger approach to quantum mechanics [1]. In quantum mechanics, in
Schrödinger picture, the time dependence of observables is encoded in the states while
the operators remain time independent. In quantum field theory using Schrödinger
picture, this implies a time-independent field operator ˙ ˆ
φ = 0. Canonical quantization
is implemented by demanding commutation relations for conjugate operators. In the
coordinate representation, position operators are represented by their eigenvalues
and momentum operators by differential equations and states by wave functions. In
field theory, we use field operators instead of position operators whose eigenvalues
are functions instead of numbers. The states are represented by wave functional and
Schrödinger equations become functional differential equations.
The techniques which are already tested in quantum mechanics can be used here.
Schrödinger equation describes the time evolution of wave functions which are the
coordinate representations of state vectors. Time-dependent Schrödinger equation
determines their unitary time evolution. Dynamical variables are treated as Hermitian
operators.
In conventional QFT, both fields and their conjugate momenta are treated as operators and used to define the Hamiltonian of the field through a Legendre transformation
of the Lagrangian. In the present treatment, fields are treated as coordinate functional
and Schrödinger equation finds an extension to obtain the functional Schrödinger
equation. Thus, QFT is basically a solution of functional Schrödinger equation.
An exact solution is usually obtained when we treat a free field but approximations are always required when we handle interacting fields. One of the simplest
approximations popularly used in quantum mechanics is the variational approximation where a trial state is expressed in terms of certain parameters which can be later
extremized to obtain bounds of relevant eigenvalues.
In functional Schrödinger picture also I employ an extension of this procedure
which is a Gaussian trial wave functional with variational parameters to implement
variational approximation. Here the study is restricted to scalar fields only but can
be extended to fermionic and gauge fields with slight modifications. Recently, great
interest in the Schrödinger picture formalism of QFT is generated in analysing QFT
in curved space-time. Nice features of the vacuum state (vacuum wave functional)
in this approach avoid several difficulties associated with QFT in curved space-time
[1–4]. The method developed can be comfortably extended to finite temperature QFT
[5]. This aspect is not discussed in this review.
The method discussed can be extended to study the de-coherence of massive fields
during inflation using variance of non-Gaussian exponent in the wave functional. It
is also used to analyse quantum-to-classical transition due to quantum de-coherence
[6, 7]. Schrödinger picture is very well suited to explore properties of the vacuum