The aim of this article is to introduce into quantum field theory, $\Bbb Z_n\times\Bbb Z_n$ graded quantum stochastic calculus with the aim of generalizing supersymmetric quantum stochastic calculus to situations where rather than just two kinds of particles, Bosons and Fermions, we can have particles of $n^2$ kinds graded by $\Bbb Z_n\times\Bbb Z_n$ rather than by $\Bbb Z_2$. Following the suggestions at the end of the book [1] by Timothy Eyre, we introduce the $\Bbb Z_n\times\Bbb Z_n$ graded tensor product by means of a bi-character constructed from a primitive $n^{th}$ root of unity and then proceed to grade both system and noise operators in Boson Fock space. This enables us to construct consistent Lie algebra supercommutation relations graded by $\Bbb Z_n\times\Bbb Z_n$ for appropriately defined graded quantum stochastic processes. Using the $\Bbb Z_n\times\Bbb Z_n$ graded quantum noise for driving the Hudson-Parthasarathy noisy Schrodinger equation combined with an appropriate non-demolition counting measurement process, we formulate the problem of $\Bbb Z_n\times\Bbb Z_n$ graded quantum filtering as a generalization of the Belavkin filter. After that, we discuss how to construct Bosonic quantum noise out of these graded quantum noise processes by tensoring with appropriately graded system operators and then how to add such noise to Bosonic field theories including quantum gravity. We also discuss quantum noise in string theories and a little bit of string field theories wherein the action for the string field is chosen based on the BRST charge quantization process that defines the condition for a state to be physical in the absence of ghosts. The addition of higher degree terms in the string field action is then based on the condition that the action should satisfy the quantum master equation for invariance of the matrix elements under the choice of the gauge fixing functional. We suggest methods by which quantum noise fields can also be taken into account in string field theories. The entire aim of adding quantum noise to a quantum field theory is motivated by the fact that the resulting noisy Hamiltonian should describe a Hudson-Parthasarathy noisy Schrodinger equation with unitary evolution on the joint System-Bath Hilbert space so that after tracing out over the bath, one obtains a quantum dynamical semigroup of TPCP maps describing system evolution alone for open quantum systems. For noisy quantum field theories, finally, we explain how to compute propagator corrections caused by noise.

QeiosAug 29, 20240 citations

CRC Press eBooks

Open Quantum Systems, Tensor Products, Fock Space Calculus, Feynman's Path Integral

Harish Parthasarathy

Published Jul 29, 2021• 0 citations

Publisher

Abstract

[ 20] The evolution of the density operator in the absence of noise.

DOI: 10.1201/9781003221432-3

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The aim of this article is to introduce into quantum field theory, $\Bbb Z_n\times\Bbb Z_n$ graded quantum stochastic calculus with the aim of generalizing supersymmetric quantum stochastic calculus to situations where rather than just two kinds of particles, Bosons and Fermions, we can have particles of $n^2$ kinds graded by $\Bbb Z_n\times\Bbb Z_n$ rather than by $\Bbb Z_2$ . Following the suggestions at the end of the book [1] by Timothy Eyre, we introduce the $\Bbb Z_n\times\Bbb Z_n$ graded tensor product by means of a bi-character constructed from a primitive $n^{th}$ root of unity and then proceed to grade both system and noise operators in Boson Fock space. This enables us to construct consistent Lie algebra supercommutation relations graded by $\Bbb Z_n\times\Bbb Z_n$ for appropriately defined graded quantum stochastic processes. Using the $\Bbb Z_n\times\Bbb Z_n$ graded quantum noise for driving the Hudson-Parthasarathy noisy Schrodinger equation combined with an appropriate non-demolition counting measurement process, we formulate the problem of $\Bbb Z_n\times\Bbb Z_n$ graded quantum filtering as a generalization of the Belavkin filter. After that, we discuss how to construct Bosonic quantum noise out of these graded quantum noise processes by tensoring with appropriately graded system operators and then how to add such noise to Bosonic field theories including quantum gravity. We also discuss quantum noise in string theories and a little bit of string field theories wherein the action for the string field is chosen based on the BRST charge quantization process that defines the condition for a state to be physical in the absence of ghosts. 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Using the $\Bbb Z_n\times\Bbb Z_n$ graded quantum noise for driving the Hudson-Parthasarathy noisy Schrodinger equation combined with an appropriate non-demolition counting measurement process, we formulate the problem of $\Bbb Z_n\times\Bbb Z_n$ graded quantum filtering as a generalization of the Belavkin filter. After that, we discuss how to construct Bosonic quantum noise out of these graded quantum noise processes by tensoring with appropriately graded system operators and then how to add such noise to Bosonic field theories including quantum gravity. We also discuss quantum noise in string theories and a little bit of string field theories wherein the action for the string field is chosen based on the BRST charge quantization process that defines the condition for a state to be physical in the absence of ghosts. 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QeiosAug 29, 20240 citations

The aim of this article is to introduce into quantum field theory,

\Bbb Z_n\times\Bbb Z_n

graded quantum stochastic calculus with the aim of generalizing supersymmetric quantum stochastic calculus to situations where rather than just two kinds of particles, Bosons and Fermions, we can have particles of

n^2

kinds graded by

\Bbb Z_n\times\Bbb Z_n

rather than by

\Bbb Z_2

. Following the suggestions at the end of the book [1] by Timothy Eyre, we introduce the

\Bbb Z_n\times\Bbb Z_n

graded tensor product by means of a bi-character constructed from a primitive

n^{th}

root of unity and then proceed to grade both system and noise operators in Boson Fock space. This enables us to construct consistent Lie algebra supercommutation relations graded by

\Bbb Z_n\times\Bbb Z_n

for appropriately defined graded quantum stochastic processes. Using the

\Bbb Z_n\times\Bbb Z_n

graded quantum noise for driving the Hudson-Parthasarathy noisy Schrodinger equation combined with an appropriate non-demolition counting measurement process, we formulate the problem of

\Bbb Z_n\times\Bbb Z_n

graded quantum filtering as a generalization of the Belavkin filter. After that, we discuss how to construct Bosonic quantum noise out of these graded quantum noise processes by tensoring with appropriately graded system operators and then how to add such noise to Bosonic field theories including quantum gravity. We also discuss quantum noise in string theories and a little bit of string field theories wherein the action for the string field is chosen based on the BRST charge quantization process that defines the condition for a state to be physical in the absence of ghosts. The addition of higher degree terms in the string field action is then based on the condition that the action should satisfy the quantum master equation for invariance of the matrix elements under the choice of the gauge fixing functional. We suggest methods by which quantum noise fields can also be taken into account in string field theories. The entire aim of adding quantum noise to a quantum field theory is motivated by the fact that the resulting noisy Hamiltonian should describe a Hudson-Parthasarathy noisy Schrodinger equation with unitary evolution on the joint System-Bath Hilbert space so that after tracing out over the bath, one obtains a quantum dynamical semigroup of TPCP maps describing system evolution alone for open quantum systems. For noisy quantum field theories, finally, we explain how to compute propagator corrections caused by noise.