Aspects of Schrödinger Picture Formalism
45
state in non-perturbative regime. The method has been applied to Yang–Mills wave
functional in (2 + 1) dimensions [8]. The relation of pre-canonical quantization of
gravity to the functional Schrödinger picture is well studied [9]. Guth and Pi, one
of the pioneers in the use of Schrödinger picture in QFT, have used it in the study
of inflationary perturbations and related the width of the Gaussian vacuum wave
functional in de-sitter space-time to Heisenberg picture scalar fields [10].
Analogous to any quantum mechanical theory (for example, path integral formalism) in functional Schrödinger picture also a kernel is a quantum mechanical
evolution operator. It represents a transition amplitude from an initial to final state.
This has been used in studying the time evolution of vacuum wave functional for QFT
on S-brane [11]. Apart from quantum field theory, the procedure is suitable for other
many-body systems in condensed matter [12] and also in (1 + 1)-dimensional nonlinear sigma model [13]. The normalized equations of motion for matter and semiclassical gravity in an inhomogeneous space-time have been obtained and Gaussian
approximation in functional Schrödinger picture has been used to analyse the time
evolution of λφ
4 model [3].
2 Scalar Field Quantization
Quantum field theories generally rely upon the concept of an action defined as S =
d
d+1
L where L is the Lagrangian density and d is the number of spatial dimensions.
The Lagrangian density for the scalar field φ(x) takes the form
L =
1
2
∂ μ φ(x)∂
μ
φ(x) − V [φ(x)] ,
where the first term represents the kinetic and the second potential terms. It is worth
noting that kinetic term has a larger invariance group than the potential since it is
invariant under the shift of the field by a global constant. φ → φ + a. Massive scalar
field Lagrangian density with mass m is clearly
L 0 =
1
2
∂ μ φ(x)∂
μ
φ(x) −
1
2
m
2
φ
2
.
This simple Lagrangian is symmetric under the discrete symmetry φ(x) → −φ(x).
If we introduce a quartic self-interacting term to obtain L 1 = L 0 −
λ
4!
φ
4 which leads
to an acceptable quantum field theory having higher order interactions like L 2 =
L 1 −
ξ
6!
φ
6 do not lead to acceptable theories, in general, but gives satisfactory theories
in lower dimensions. We can define the canonical momentum operator
(x) =
∂L
∂(∂ 0 φ)
= ∂ 0 φ(x).
45
state in non-perturbative regime. The method has been applied to Yang–Mills wave
functional in (2 + 1) dimensions [8]. The relation of pre-canonical quantization of
gravity to the functional Schrödinger picture is well studied [9]. Guth and Pi, one
of the pioneers in the use of Schrödinger picture in QFT, have used it in the study
of inflationary perturbations and related the width of the Gaussian vacuum wave
functional in de-sitter space-time to Heisenberg picture scalar fields [10].
Analogous to any quantum mechanical theory (for example, path integral formalism) in functional Schrödinger picture also a kernel is a quantum mechanical
evolution operator. It represents a transition amplitude from an initial to final state.
This has been used in studying the time evolution of vacuum wave functional for QFT
on S-brane [11]. Apart from quantum field theory, the procedure is suitable for other
many-body systems in condensed matter [12] and also in (1 + 1)-dimensional nonlinear sigma model [13]. The normalized equations of motion for matter and semiclassical gravity in an inhomogeneous space-time have been obtained and Gaussian
approximation in functional Schrödinger picture has been used to analyse the time
evolution of λφ
4 model [3].
2 Scalar Field Quantization
Quantum field theories generally rely upon the concept of an action defined as S =
d
d+1
L where L is the Lagrangian density and d is the number of spatial dimensions.
The Lagrangian density for the scalar field φ(x) takes the form
L =
1
2
∂ μ φ(x)∂
μ
φ(x) − V [φ(x)] ,
where the first term represents the kinetic and the second potential terms. It is worth
noting that kinetic term has a larger invariance group than the potential since it is
invariant under the shift of the field by a global constant. φ → φ + a. Massive scalar
field Lagrangian density with mass m is clearly
L 0 =
1
2
∂ μ φ(x)∂
μ
φ(x) −
1
2
m
2
φ
2
.
This simple Lagrangian is symmetric under the discrete symmetry φ(x) → −φ(x).
If we introduce a quartic self-interacting term to obtain L 1 = L 0 −
λ
4!
φ
4 which leads
to an acceptable quantum field theory having higher order interactions like L 2 =
L 1 −
ξ
6!
φ
6 do not lead to acceptable theories, in general, but gives satisfactory theories
in lower dimensions. We can define the canonical momentum operator
(x) =
∂L
∂(∂ 0 φ)
= ∂ 0 φ(x).