We analyze an example of a photon in a superposition of different modes, and ask what is the degree of their entanglement with a vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for the field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in the Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the $\ensuremath{\infty}$-representation and show in two ways that in two-mode states the modes are maximally entangled with the vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with the vacuum is present even for single-mode states. Still, the degree of entanglement here is difficult to estimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and large.We analyze an example of a photon in a superposition of different modes, and ask what is the degree of their entanglement with a vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for the field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in the Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the $\ensuremath{\infty}$-representation and show in two ways that in two-mode states the modes are maximally entangled with the vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with the vacuum is present even for single-mode states. Still, the degree of entanglement here is difficult to estimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and large.

arXiv:quant-ph/0512253Physical Review AApr 21, 200624 citations

Stochastic metric space and quantum mechanics

Y. Kurihara

A new idea for the quantization of dynamic systems, as well as space time itself, using a stochastic metric is proposed. The quantum mechanics of a mass point is constructed on a space time manifold using a stochastic metric. A stochastic metric space is, in brief, a metric space whose metric tensor is given stochastically according to some appropriate distribution function. A mathematically consistent model of a space time manifold equipping a stochastic metric is proposed in this work. The quantum theory in the local Minkowski space can be recognized as a classical theory on the stochastic Lorentz-metric-space. A stochastic calculus on the space time manifold is performed using white noise functional analysis. A path-integral quantization is introduced as a stochastic integration of a function of the action integral, and it is shown that path-integrals on the stochastic metric space are mathematically well-defined for large variety of potential functions. The Newton–Nelson equation of motion can also be obtained from the Newtonian equation of motion on the stochastic metric space. It is also shown that the commutation relation required under the canonical quantization is consistent with the stochastic quantization introduced in this work. The quantum effects of general relativity are also analyzed through natural use of the stochastic metrics. Some example of quantum effects on the Universe is discussed.A new idea for the quantization of dynamic systems, as well as space time itself, using a stochastic metric is proposed. The quantum mechanics of a mass point is constructed on a space time manifold using a stochastic metric. A stochastic metric space is, in brief, a metric space whose metric tensor is given stochastically according to some appropriate distribution function. A mathematically consistent model of a space time manifold equipping a stochastic metric is proposed in this work. The quantum theory in the local Minkowski space can be recognized as a classical theory on the stochastic Lorentz-metric-space. A stochastic calculus on the space time manifold is performed using white noise functional analysis. A path-integral quantization is introduced as a stochastic integration of a function of the action integral, and it is shown that path-integrals on the stochastic metric space are mathematically well-defined for large variety of potential functions. The Newton–Nelson equation of motion can also be obtained from the Newtonian equation of motion on the stochastic metric space. It is also shown that the commutation relation required under the canonical quantization is consistent with the stochastic quantization introduced in this work. The quantum effects of general relativity are also analyzed through natural use of the stochastic metrics. Some example of quantum effects on the Universe is discussed.

arXiv:1612.04228Journal of Physics CommunicationsJan 17, 201810 citations

Remarks on connections between quantum mechanics and stochastic quantization

道夫西岡

Oct 1, 19820 citations

N+2

subsystems, with

N

unspecified and large.