Spontaneous symmetry breaking in the problem of scattering a particle by a one-dimensional $\delta$-potential barrier

Nikolay Leonidovich Chuprikov

Published May 3, 2018• 0 citations

Abstract

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Spontaneous symmetry breaking in the problem of scattering a particle by a one-dimensional $\delta$ -potential barrierSpontaneous symmetry breaking in the problem of scattering a particle by a one-dimensional $\delta$-potential barrier

Nikolay Leonidovich Chuprikov

Published May 3, 2018• 0 citations

OpenAlex

Abstract

P

- and

T

-symmetries and the appearance of a superselection rule. Here, the non-unitary scattering state is a mixed vector state, which is a coherent superposition of transmission and reflection states related to different Hamiltonians. Superselection operators for the transmission and reflection subprocesses are defined.

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On the need to revise the superposition principle for a particle with a continuous energy spectrum: superselection rule for a one-dimensional scattering

Nikolay Leonidovich Chuprikov

It is shown that spontaneous symmetry breaking occurs in this scattering problem. As a result, the unitary, asymptotically free quantum dynamics of this process is unstable. It transforms into dynamics associated with the violation of $P$ - and $T$ -symmetries and the appearance of a superselection rule. Here, the non-unitary scattering state is a mixed vector state, which is a coherent superposition of transmission and reflection states related to different Hamiltonians. Superselection operators for the transmission and reflection subprocesses are defined.It is shown that spontaneous symmetry breaking occurs in this scattering problem. As a result, the unitary, asymptotically free quantum dynamics of this process is unstable. It transforms into dynamics associated with the violation of $P$- and $T$-symmetries and the appearance of a superselection rule. Here, the non-unitary scattering state is a mixed vector state, which is a coherent superposition of transmission and reflection states related to different Hamiltonians. Superselection operators for the transmission and reflection subprocesses are defined.

arXiv:1805.03952May 3, 20180 citations

Scattering a particle on a one-dimensional $\delta$ -potential barrier: asymptotic superselection ruleScattering a particle on a one-dimensional $\delta$-potential barrier: asymptotic superselection rule

Nikolay Leonidovich Chuprikov

As is stated, the modern mathematical ( $C^*$ -algebraic) scattering theory for the formal Hamiltonian with a one-dimensional short-range potential, developed on the basis of the operational definition of observables, describes unitary asymptotically free dynamics. But, as it s shown in this paper by the example of the $\delta$ -potential, this Hamiltonian can describe either unitary asymptotically not free dynamics (endless interaction process) or non-unitary asymptotically free dynamics (scattering process). In the case of scattering, the unboundedness of the position operator plays a key role. Each solution to the Schrodinger equation, describing the scattering process with a one-sided incidence of a particle on the barrier, is non-unique in the limit $t\to \infty$ , when it is a superposition of unconnected states (left and right out-asymptotes) localized in the disjoint spatial regions located on opposite sides of the barrier --- it describes non-unitary quantum dynamics. Measurement of the coordinates of the transmitted and reflected particles using one experimental setup of finite dimensions, as is assumed in the operational approach, is impossible. We need two such setups --- one for the transmitted particles, and the other for the reflected ones. Thus, such a solution is a mixed vector state --- this process is governed by the asymptotic superselection rule.As is stated, the modern mathematical ($C^*$-algebraic) scattering theory for the formal Hamiltonian with a one-dimensional short-range potential, developed on the basis of the operational definition of observables, describes unitary asymptotically free dynamics. But, as it s shown in this paper by the example of the $\delta$-potential, this Hamiltonian can describe either unitary asymptotically not free dynamics (endless interaction process) or non-unitary asymptotically free dynamics (scattering process). In the case of scattering, the unboundedness of the position operator plays a key role. Each solution to the Schrodinger equation, describing the scattering process with a one-sided incidence of a particle on the barrier, is non-unique in the limit $t\to \infty$, when it is a superposition of unconnected states (left and right out-asymptotes) localized in the disjoint spatial regions located on opposite sides of the barrier --- it describes non-unitary quantum dynamics. Measurement of the coordinates of the transmitted and reflected particles using one experimental setup of finite dimensions, as is assumed in the operational approach, is impossible. We need two such setups --- one for the transmitted particles, and the other for the reflected ones. Thus, such a solution is a mixed vector state --- this process is governed by the asymptotic superselection rule.

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It is shown that non-stationary solutions of the Schrödinger equation, which describes the quantum dynamics of a particle in the field of a one-dimensional delta potential (1DDP), are divided into two classes: some define pure states that have no free dynamics as $t\to\mp\infty$ ; others define states with asymptotically free dynamics but represent mixed states in whose space the asymptotic superselection rule holds. That is, according to the Schrödinger equation, {\it pure scattering} states predicted by the conventional model of this scattering process do not exist. On mixed scattering states, the Hamiltonian with 1DDP is defined only in superselection sectors. The scattering process with one-sided incidence of a particle on 1DDP represents a decoherence process in a closed system.It is shown that non-stationary solutions of the Schrödinger equation, which describes the quantum dynamics of a particle in the field of a one-dimensional delta potential (1DDP), are divided into two classes: some define pure states that have no free dynamics as $t\to\mp\infty$; others define states with asymptotically free dynamics but represent mixed states in whose space the asymptotic superselection rule holds. That is, according to the Schrödinger equation, {\it pure scattering} states predicted by the conventional model of this scattering process do not exist. On mixed scattering states, the Hamiltonian with 1DDP is defined only in superselection sectors. The scattering process with one-sided incidence of a particle on 1DDP represents a decoherence process in a closed system.

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Spontaneous symmetry breaking in the non-linear Schrödinger hierarchy with defect

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We introduce and solve the one-dimensional quantum non-linear Schrödinger (NLS) equation for an N -component field defined on the real line with a defect sitting at the origin. The quantum solution is constructed using the quantum inverse scattering method based on the concept of reflection–transmission (RT) algebras recently introduced. The symmetry of the model is generated by the reflection and transmission defect generators defining a defect subalgebra. We classify all the corresponding reflection and transmission matrices. This provides the possible boundary conditions obeyed by the canonical field and we compute these boundary conditions explicitly. Finally, we exhibit a phenomenon of spontaneous symmetry breaking induced by the defect and identify the unbroken generators as well as the exact remaining symmetry.We introduce and solve the one-dimensional quantum non-linear Schrödinger (NLS) equation for an N -component field defined on the real line with a defect sitting at the origin. The quantum solution is constructed using the quantum inverse scattering method based on the concept of reflection–transmission (RT) algebras recently introduced. The symmetry of the model is generated by the reflection and transmission defect generators defining a defect subalgebra. We classify all the corresponding reflection and transmission matrices. This provides the possible boundary conditions obeyed by the canonical field and we compute these boundary conditions explicitly. Finally, we exhibit a phenomenon of spontaneous symmetry breaking induced by the defect and identify the unbroken generators as well as the exact remaining symmetry.

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Nikolay Leonidovich Chuprikov

It is shown that in the state space of a quantum particle involved in a one-dimensional completed scattering (OCS) there is a superselection rule (SSR) induced by the dichotomous physical context that determines the properties of a quantum ensemble of scattering particles in the disjoint spatial regions located on the different sides of the potential barrier at asymptotically large distances from it. At the initial stage of the OCS this context is associated, in the general case, with two sources of particles located on the different sides of the barrier; and at the final stage it is associated, in the general case, with two particle detectors located on different sides of the barrier. The role of a superselection operator is played by the Pauli matrix $σ_3$ that divides the space of in- and out-asymptotes into two coherent sectors. In the course of the OCS, its unitary quantum dynamics crosses these sectors. In particular, in the scattering problem with one source and two detectors, a pure initial state of a particle is converted into a mixed final state. According to this rule, the OCS is a complex quantum process consisting of two coherent subprocesses -- transmission and reflection; the average value of any observable as well as characteristic times can be defined for the subprocesses only. It is shown that the quantum dynamics of both subprocesses at all stages of scattering is uniquely determined by the initial state of the whole ensemble of particles and by the final states of transmitted and reflected particles.It is shown that in the state space of a quantum particle involved in a one-dimensional completed scattering (OCS) there is a superselection rule (SSR) induced by the dichotomous physical context that determines the properties of a quantum ensemble of scattering particles in the disjoint spatial regions located on the different sides of the potential barrier at asymptotically large distances from it. At the initial stage of the OCS this context is associated, in the general case, with two sources of particles located on the different sides of the barrier; and at the final stage it is associated, in the general case, with two particle detectors located on different sides of the barrier. The role of a superselection operator is played by the Pauli matrix $σ_3$ that divides the space of in- and out-asymptotes into two coherent sectors. In the course of the OCS, its unitary quantum dynamics crosses these sectors. In particular, in the scattering problem with one source and two detectors, a pure initial state of a particle is converted into a mixed final state. According to this rule, the OCS is a complex quantum process consisting of two coherent subprocesses -- transmission and reflection; the average value of any observable as well as characteristic times can be defined for the subprocesses only. It is shown that the quantum dynamics of both subprocesses at all stages of scattering is uniquely determined by the initial state of the whole ensemble of particles and by the final states of transmitted and reflected particles.

arXiv:1412.7657Dec 24, 20140 citations

Spontaneous symmetry breaking in the problem of scattering a particle by a one-dimensional

\delta

-potential barrier

P

- and

T

P

- and

T

Scattering a particle on a one-dimensional

\delta

-potential barrier: asymptotic superselection rule

As is stated, the modern mathematical (

C^*

-algebraic) scattering theory for the formal Hamiltonian with a one-dimensional short-range potential, developed on the basis of the operational definition of observables, describes unitary asymptotically free dynamics. But, as it s shown in this paper by the example of the

\delta

-potential, this Hamiltonian can describe either unitary asymptotically not free dynamics (endless interaction process) or non-unitary asymptotically free dynamics (scattering process). In the case of scattering, the unboundedness of the position operator plays a key role. Each solution to the Schrodinger equation, describing the scattering process with a one-sided incidence of a particle on the barrier, is non-unique in the limit

t\to \infty

, when it is a superposition of unconnected states (left and right out-asymptotes) localized in the disjoint spatial regions located on opposite sides of the barrier --- it describes non-unitary quantum dynamics. Measurement of the coordinates of the transmitted and reflected particles using one experimental setup of finite dimensions, as is assumed in the operational approach, is impossible. We need two such setups --- one for the transmitted particles, and the other for the reflected ones. Thus, such a solution is a mixed vector state --- this process is governed by the asymptotic superselection rule.

Asymptotes of non-stationary solutions of the Schrödinger equation for a particle interacting with a one-dimensional

δ

-potential

t\to\mp\infty

; others define states with asymptotically free dynamics but represent mixed states in whose space the asymptotic superselection rule holds. That is, according to the Schrödinger equation, {\it pure scattering} states predicted by the conventional model of this scattering process do not exist. On mixed scattering states, the Hamiltonian with 1DDP is defined only in superselection sectors. The scattering process with one-sided incidence of a particle on 1DDP represents a decoherence process in a closed system.

σ_3

and the role of superselection (coherent) sectors is played by the above subspaces. By the SSR any superposition of states from different coherent sectors is a mixed state, and splitting the incident wave packet into the transmitted and reflected parts is nothing but a conversion of a pure state into a mixed one. The average values of any observable can be defined only for the transmission and reflection subprocesses. The former evolves within a single coherent sector of the Hilbert space, in the momentum representation; while the latter evolves within a single coherent sector of this space, in the coordinate representation.

σ_3

that divides the space of in- and out-asymptotes into two coherent sectors. In the course of the OCS, its unitary quantum dynamics crosses these sectors. In particular, in the scattering problem with one source and two detectors, a pure initial state of a particle is converted into a mixed final state. According to this rule, the OCS is a complex quantum process consisting of two coherent subprocesses -- transmission and reflection; the average value of any observable as well as characteristic times can be defined for the subprocesses only. It is shown that the quantum dynamics of both subprocesses at all stages of scattering is uniquely determined by the initial state of the whole ensemble of particles and by the final states of transmitted and reflected particles.