Aspects of Schrödinger Picture
Formalism
K. P. Satheesh
Abstract Application of functional Schrödinger picture formalism to scalar theories
is described at an elementary level. The basic formalism is applied to a free scalar
field. To handle interacting fields, variational approximation with Gaussian wave
functional as trial wave functional is developed. The procedure is applied to φ
4 and
φ
6 self-interacting scalar fields. The effective action, effective potential and effective
mass are calculated up to two-loop and three-loop levels for φ
4 and φ
6 , respectively.
Calculation of static effective potential is done and compared with Gaussian effective
potential formalism and CJT formalism.
Keywords Schrödinger picture · Quantum field theory · De-coherence · Gaussian
effective potential
1 Introduction
Schrödinger picture formalism in quantum field theory is a generalization from ordinary quantum mechanics to the infinite number of degrees of freedom which comprises a field. Quantum state in this approach is a functional superposition of field configurations. In studying the detailed structures of both bosonic and fermionic quantum
fields, the variational approximation applied to functional Schrödinger picture has
been found to be very useful. Here I use the simplest variational approximation which
is a Gaussian approximation and is found to provide better information than largeN approximation. It is found that the effective potential expression obtained using
Schrödinger picture and Cornwall–Jackiw–Tomboulis formalism opens a channel
towards finite temperature field theories. Vacuum properties in this formalism are
conceptually simpler than Fock Space vacuum (normal ordering not required) and
find several applications in quantum fields in curved space-time.
K. P. Satheesh (B)
Government Brennen College, Thalassery, Kerala, India
e-mail: satheeshsithara@gmail.com
Institute for Intensive Research in Basic Sciences, M.G University, Kottayam, Kerala, India
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_4
43
Formalism
K. P. Satheesh
Abstract Application of functional Schrödinger picture formalism to scalar theories
is described at an elementary level. The basic formalism is applied to a free scalar
field. To handle interacting fields, variational approximation with Gaussian wave
functional as trial wave functional is developed. The procedure is applied to φ
4 and
φ
6 self-interacting scalar fields. The effective action, effective potential and effective
mass are calculated up to two-loop and three-loop levels for φ
4 and φ
6 , respectively.
Calculation of static effective potential is done and compared with Gaussian effective
potential formalism and CJT formalism.
Keywords Schrödinger picture · Quantum field theory · De-coherence · Gaussian
effective potential
1 Introduction
Schrödinger picture formalism in quantum field theory is a generalization from ordinary quantum mechanics to the infinite number of degrees of freedom which comprises a field. Quantum state in this approach is a functional superposition of field configurations. In studying the detailed structures of both bosonic and fermionic quantum
fields, the variational approximation applied to functional Schrödinger picture has
been found to be very useful. Here I use the simplest variational approximation which
is a Gaussian approximation and is found to provide better information than largeN approximation. It is found that the effective potential expression obtained using
Schrödinger picture and Cornwall–Jackiw–Tomboulis formalism opens a channel
towards finite temperature field theories. Vacuum properties in this formalism are
conceptually simpler than Fock Space vacuum (normal ordering not required) and
find several applications in quantum fields in curved space-time.
K. P. Satheesh (B)
Government Brennen College, Thalassery, Kerala, India
e-mail: satheeshsithara@gmail.com
Institute for Intensive Research in Basic Sciences, M.G University, Kottayam, Kerala, India
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_4
43