JMP #58, 4: path integrals and friends

The path integral formulation of quantum mechanics replaces the single, classical trajectory of a system with the sum over an infinity of quantum possible trajectories. To compute this sum is used a functional integral. The most famous interpretation is dued by Richard Feynman. In an Euclidean spacetime we speak about Euclidean path integral:

Bernardo, R. C. S., & Esguerra, J. P. H. (2017). Euclidean path integral formalism in deformed space with minimum measurable length. Journal of Mathematical Physics, 58(4), 042103. doi:10.1063/1.4979797

We study time-evolution at the quantum level by developing the Euclidean path-integral approach for the general case where there exists a minimum measurable length. We derive an expression for the momentum-space propagator which turns out to be consistent with recently developed $\beta$-canonical transformation. We also construct the propagator for maximal localization which corresponds to the amplitude that a state which is maximally localized at location $\xi'$ propagates to a state which is maximally localized at location $\xi"$ in a given time. Our expression for the momentum-space propagator and the propagator for maximal localization is valid for any form of time-independent Hamiltonian. The nonrelativistic free particle, particle in a linear potential, and the harmonic oscillator are discussed as examples.

Other papers from JMP #58, 4, follows:

JMP 58, 3: math paradoxes in quantum mechanics

Facchi, P., & Ligabò, M. (2017). Large-time limit of the quantum Zeno effect Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4978851 (arXiv)

If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, $t \propto N^\alpha$. The limit survival probability exhibits a sudden jump from $1$ to $0$ at $\alpha = 1/2$, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for $\alpha \geq 1$, the limit probability becomes sensitive to the spectral properties of the initial state and to the arithmetic properties of the measurement periods.

Selvitella, A. (2017). The Simpson’s paradox in quantum mechanics Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4977784 (sci-hub)

In probability and statistics, the Simpson’s paradox is a paradox in which a trend that appears in different groups of data disappears when these groups are combined, while the reverse trend appears for the aggregate data. In this paper, we give some results about the occurrence of the Simpson’s paradox in quantum mechanics. In particular, we prove that the Simpson’s paradox occurs for solutions of the quantum harmonic oscillator both in the stationary case and in the non-stationary case. In the non-stationary case, the Simpson’s paradox is persistent: if it occurs at any time $t=\tilde t$, then it occurs at any time $t\not= \tilde t$. Moreover, we prove that the Simpson’s paradox is not an isolated phenomenon, namely, that, close to initial data for which it occurs, there are lots of initial data (a open neighborhood), for which it still occurs. Differently from the case of the quantum harmonic oscillator, we also prove that the paradox appears (asymptotically) in the context of the nonlinear Schrödinger equation but at intermittent times.

Read also: Two quantum Simpson's paradoxes

Meng, F., & Liu, C. (2017). Necessary and sufficient conditions for the existence of time-dependent global attractor and application Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4978329 (sci-hub)

In this paper, we are concerned with infinite dimensional dynamical systems in time-dependent space. First, we characterize some necessary and sufficient conditions for the existence of the time-dependent global attractor by using a measure of noncompactness. Then, we give a new method to verify the sufficient condition. As a simple application, we prove the existence of the time-dependent global attractor for the damped equation in strong topological space.

Cen, J., Correa, F., & Fring, A. (2017). Time-delay and reality conditions for complex solitons Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4978864 (arXiv)

We compute lateral displacements and time-delays for scattering processes of complex multi-soliton solutions of the Korteweg de-Vries equation. The resulting expressions are employed to explain the precise distinction between solutions obtained from different techniques, Hirota’s direct method and a superposition principle based on Bäcklund transformations. Moreover they explain the internal structures of degenerate compound multi-solitons previously constructed. Their individual one-soliton constituents are time-delayed when scattered amongst each other. We present generic formulae for these time-dependent displacements. By recalling Gardner’s transformation method for conserved charges, we argue that the structure of the asymptotic behaviour resulting from the integrability of the model together with its $PT$-symmetry ensures the reality of all of these charges, including in particular the mass, the momentum, and the energy.

Wilming, H., Kastoryano, M., Werner, A., & Eisert, J. (2017). Emergence of spontaneous symmetry breaking in dissipative lattice systems Journal of Mathematical Physics, 58 (3) DOI: 10.1063/1.4978328 (arXiv)

A cornerstone of the theory of phase transitions is the observation that many-body systems exhibiting a spontaneous symmetry breaking in the thermodynamic limit generally show extensive fluctuations of an order parameter in large but finite systems. In this work, we introduce the dynamical analog of such a theory. Specifically, we consider local dissipative dynamics preparing an equilibrium steady-state of quantum spins on a lattice exhibiting a discrete or continuous symmetry but with extensive fluctuations in a local order parameter. We show that for all such processes, there exist asymptotically stationary symmetry-breaking states, i.e., states that become stationary in the thermodynamic limit and give a finite value to the order parameter. We give results both for discrete and continuous symmetries and explicitly show how to construct the symmetry-breaking states. Our results show in a simple way that, in large systems, local dissipative dynamics satisfying detailed balance cannot uniquely and efficiently prepare states with extensive fluctuations with respect to local operators. We discuss the implications of our results for quantum simulators and dissipative state preparation.

JMP 58, 2: quantum abstract

I'm returning with the selections from the Journal of Mathematical Physics. I'm arranging two further post for this series for the next weeks. Stay tuned!

Alhaidari, A., & Taiwo, T. (2017). Wilson-Racah quantum system Journal of Mathematical Physics, 58 (2) DOI: 10.1063/1.4975138 (arXiv)

Using a recent formulation of quantum mechanics without a potential function, we present a four-parameter system associated with the Wilson and Racah polynomials. The continuum scattering states are written in terms of the Wilson polynomials whose asymptotics give the scattering amplitude and phase shift. On the other hand, the finite number of discrete bound states are associated with the Racah polynomials.

Dorsch, F. (2017). Accumulation rate of bound states of dipoles generated by point charges in strained graphene Journal of Mathematical Physics, 58 (2) DOI: 10.1063/1.4976201 (arXiv)

We consider strained graphene, modelled by the two-dimensional massive Dirac operator, with potentials corresponding to charge distributions with vanishing total charge, non-vanishing dipole moment and finitely many point charges of subcritical coupling constants located in the graphene sheet. We show that the bound state energies accumulate exponentially fast at the edges of the spectral gap by determining the leading order of the accumulation rate.

JMP 58, 1: magnetic monopoles, spacetime and gravity

Just another selection of papers from the Journal of Mathematical Physics. I would start with the folowing paper:

Fine, D., & Sawin, S. (2017). Path integrals, supersymmetric quantum mechanics, and the Atiyah-Singer index theorem for twisted Dirac Journal of Mathematical Physics, 58 (1) DOI: 10.1063/1.4973368

Feynman’s time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure that these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of $N = 1/2$ supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.

Kováčik, S., & Prešnajder, P. (2017). Magnetic monopoles in noncommutative quantum mechanics Journal of Mathematical Physics, 58 (1) DOI: 10.1063/1.4973503

We discuss a certain generalization of the Hilbert space of states in noncommutative quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).

JMP 57, 12: quantum mechanics for the universe

I will begin to publish posts in a more continuous way, also recovering some drafts that I'm not able to conclude in 2016. For now I propose you a selection of articles from the Journal of Mathematical Physics, vol.57, issue 12:

Andersson, A. (2016). Electromagnetism in terms of quantum measurements Journal of Mathematical Physics, 57 (12) DOI: 10.1063/1.4972287

We consider the question whether electromagnetism can be derived from the theory of quantum measurements. It turns out that this is possible, both for quantum and classical electromagnetism, if we use more recent innovations such as smearing of observables and simultaneous measurability. In this way, we justify the use of von Neumann-type measurement models for physical processes. We apply the operational quantum measurement theory to gain insight into fundamental aspects of quantum physics. Interactions of von Neumann type make the Heisenberg evolution of observables describable using explicit operator deformations. In this way, one can obtain quantized electromagnetism as a measurement of a system by another. The relevant deformations (Rieffel deformations) have a mathematically well-defined "classical" limit which is indeed classical electromagnetism for our choice of interaction.

Aerts, D., & Sassoli de Bianchi, M. (2016). The extended Bloch representation of quantum mechanics: Explaining superposition, interference, and entanglement Journal of Mathematical Physics, 57 (12) DOI: 10.1063/1.4973356

An extended Bloch representation of quantum mechanics was recently derived to offer a possible (hidden-measurements) solution to the measurement problem. In this article we use this representation to investigate the geometry of superposition and entangled states, explaining interference effects and entanglement correlations in terms of the different orientations a state-vector can take within the generalized Bloch sphere. We also introduce a tensorial determination of the generators of $SU(N)$, which we show to be particularly suitable for the description of multipartite systems, from the viewpoint of the sub-entities. We then use it to show that non-product states admit a general description where sub-entities can remain in well-defined states, even when entangled. This means that the completed version of quantum mechanics provided by the extended Bloch representation, where density operators are also considered to be representative of genuine states (providing a complete description), not only offers a plausible solution to the measurement problem but also to the lesser-known entanglement problem. This is because we no longer need to give up the general physical principle saying that a composite entity exists and therefore is in a well-defined state, if and only if its components also exist and therefore are also in well-defined states.