Wigner's theorem

The Wigner’s theorem was formulated and demonstrated for the first time by Eugene Paul Wigner on Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum(1). It states that for each symmetry transformation in Hilbert’s space there exists a unitary or anti-unitary operator, uniquely determined less than a phase factor.
For symmetry transformation, we intend a space transformation that preserved the characteristics of a given physical system. Asymmetry transformation implies also a change of reference system.
Invariants
Invariants play a key role in physics, being the quantities that, in any reference system, are unchanged. With the advent of quantum physics, their importance increased, particularly in the formulation of a relativistic quantum field theory. One of the most important tools in the study of invariants is the Wigner’s theorem, an instrument of fundamental importance for all the development of quantum theory.
In particular, Wigner was interested in determining the properties of transformations that preserve the transition’s probability between two different quantum states. Given $\phi$ the wave function detected by the first observer, and $\bar {\phi}$ the wave function detected by the second observer, Wigner assumed that the equality \[|\langle \psi | \phi \rangle| = |\langle \bar \psi | \bar \phi \rangle|\] must be valid for all $\psi$ and $\phi$.
In the end, if we exclude time inversions, we find that the operator $\operatorname{O}_{R}$, such that $\bar{\phi} = \operatorname{O} _{R} \phi$, must be unitary and linear, but also anti-unitary and anti-linear. Consequence of this fact is that the two observers’ descriptions are equivalent. So the first observes $\phi$, the second $\bar{\phi}$, while the operator $\operatorname{H}$ for the first will be $\operatorname{O}_R \operatorname{H} \operatorname{O}_R^{-1}$ for the second.

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:

The quantum Zeno paradox

The standard axioms of quantum mechanics imply that in the limit of continuous observation a quantum system cannot evolve.
(Andrew Hodges in Alan Turing: the logical and physical basis of computing - pdf)
Initially known as Turing’s paradox, in honor of the mathematician who formulated it in the 1950s, was subsequently identified as quantum Zeno effect, resulting an advanced version of the famous Zeno’s arrow paradox, whose phylosophical result is the negation of motion. A first formulation and derivation of the effect is found in Does the lifetime of an unstable system depend on the measuring apparatus?(1), while George Sudarshan and Baidyanath Misra were the first to identify it as quantum Zeno paradox. The two theoretical physicists established that an unstable particle will not decay as long as it is kept under continuous observation(2). However they try to save goat and cabbage:
There is a fundamental principle in quantum theory that denies the possibility of continuous observation.(2)
On the other hand, Ghirardi, Omero, Weber and Rimini show that:
if the uncertainty relations are properly taken into account the arguments leading to the paradox are not valid.(3)

Adiabatic pendulum

In physics quantities which change little under slow changes of parameter, are called adiabatic invariants(1). The pendulum could be an adiabatic invariants. To get it, as well as to get any of the adiabatic invariants, the person changing the parameters of the system must not see what state the system is in(1).

  1. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag (1989)

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.

There's no two without three

On January 4th, 2017, LIGO detected two black holes merging into one. One of the black holes was 32 times the mass of the Sun, while the other was 19 times the mass of the Sun. When they merged, they created a black hole 49 times the mass of the Sun. The coalescence instantly converted 2 solar masses of black hole mass into the energy that rattled spacetime enough to generate the gravitational waves we detected almost 3 billion years after it occurred. (Caltech/MIT/LIGO Lab)
Read also: LIGO Picks Up on the Third Ring
LIGO Scientific and Virgo Collaboration (2017). GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2 Physical Review Letters, 118 (22) DOI: 10.1103/PhysRevLett.118.221101

Buzz Aldrin: from Milano to Mars

Yesterday Buzz Aldrin was in Milano for the Wired Next Fest. The conference room was full of people, and I cannot enter in it. So I see Buzz's conference from the large screen outside in front of the stage. Even if mediated in this way, I could perceive the power of the character, the charisma, the magnetism of a man who goes by now for the nineties but who is not tired yet to point his eyes to the sky. Towards Mars, for precision!

Holistic Detective Agency

Have you ever needed a holistic detective agency? And do you know what a holistic detective agency is? If you do not know, don't panic: I am going to say what it is this strange agency.
The term ‘holistic’ refers to my [detective's] conviction that what we are concerned with here is the fundamental interconnectedness of all things. I do not concern myself with such petty things as fingerprint powder, telltale pieces of pocket fluff and inane footprints. I see the solution to each problem as being detectable in the pattern and web of the whole. The connections between causes and effects are often much more subtle and complex than we with our rough and ready understanding of the physical world might naturally suppose.
And what are its secret origins?
Well, some researchers were once conducting such an experiment [Schroedinger's cat], but when they opened up the box, the cat was neither alive nor dead but was in fact completely missing, and they called me in to investigate. I was able to deduce that nothing very dramatic had happened. The cat had merely got fed up with being repeatedly locked up in a box and occasionally gassed and had taken the first opportunity to hoof it through the window. It was for me the work of a moment to set a saucer of milk by the window and call “Bernice” in an enticing voice -- the cat’s name was Bernice, you understand -- and the cat was soon restored. A simple enough matter, but it seemed to create quite an impression in certain circles, and soon one thing led to another as they do and it all culminated in the thriving career you see before you.

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.

Physics Education vol.52: Ciênsação and others educational papers

One of the most interesting paper published on the last issue of Physics Education is Ciênsação: gaining a feeling for sciences about a learning repository for high school theachers.
The project born for brazilian schools, but I think that it could be useful for allo teachers in the world. For example in most educational systems there are the same reasons against the introduction of hands-on experiments in classroom:
  • lack of time;
  • insecurity and lack of training;
  • lack of resources and infrastructure.
So the repository Ciênsação propose to all teachers some interesting and simple experiments. It try
(...) to convince teachers that short experiments — which may take just a couple of minutes or even seconds to conduct — can smoothly transit to productive class discussions, in which students simultaneously advance their fact knowledge, deepen their understanding and foster their scientific skills.
If we see some experiments (for example the brief activity about magnetism), we can apreciate the simple integration of the experiment in the usual lesson with few materials.
I think that the phylosophy of Ciênsação is very near to my (past) teaching activities:
The research tasks, around which Ciênsação experiments are built, invite students to actively do science, instead of merely reproducing known results and confirming textbook claims. Giving students a few minutes to pursue such a task autonomously in small groups, allows them not only to experience the excitement of discovery, but also to experience science as a creative activity, as a craft they can master, rather than the privilege of an elite called 'scientists'.
For theachers that are intrested to sbmit their teaching activities, there's also a submission form.
Henrique Abreu de Oliveira, M., & Fischer, R. (2017). Ciênsação: gaining a feeling for sciences Physics Education, 52 (2) DOI: 10.1088/1361-6552/aa5430
And now some others interesting papers:

Feynman in comics

with @estuan about #Feynman #Ottaviani #Myrick #comics #physics
Italian version written with Maria-Angela Silleni.
Feynman by Ottaviani and Myrick, reported by Maria Popova as one of the top 11 scientific popular books of 2012, is a splendid example of how to make interesting science. Even with the comics.
Telling a man’s life is always an arduous and difficult task. So it is necessary to make choices, often focusing on successes and leaving aside failures, especially if you talk about a physicist who has been interested in the most disparate fields within his specialization.
This is the some idea that Lawrence Krauss has dealt with the life of Richard Feynman in Quantum Man and the same spirit seems to animate Feynman, the graphic novel by Jim Ottaviani and Leland Myrick.
Introducing Feynman?
After the series of Introducing, a hybrid of illustrated and comic books about science, one of the most effective comics dedicated to science, brings the signature of Ottaviani, which, on the pages of Suspended In Language, with the contribution of the comic artist Leland Purvis, told the life of Niels Bohr, the master of the Copenhagen School, whose interpretation of quantum mechanics dominated during the first steps of this new approach of physics to nature.
Ottaviani knows very well the risks in comic book science genre: in particular, falling into teaching and slamming the reader into boredom is always around the corner; so thanks to an episode narrative (which is also a limit, as we shall see below), accompanied by synthetic drawings, the volume reaches the disclosure purpose without betraying the aspect of entertainment.
A great role in the success of the graphic novel is dued by the subject: Feynman, Nobel Prize in Physics in 1965, was an eccentric character with a strong sense of humor and impatient with social conventions. Thanks to these features he often found himself in embarrassing situations with unexpected consequences. A passionate bongo player and amateur sketcher of naked women, picked up his two-volume adventures, Surely You’re Joking, Mr. Feynman! and What Do You Care What Other People Think?, which wrote to seem not overly serious: and most of the episodes narrated in the comic are getting from his autobiographical stories.

1st Italian Astrostatistics School

Astrostatistics is a discipline which spans astrophysics, statistical analysis and data mining. It is used to process the vast amount of data produced by automated scanning of the cosmos, to characterize complex datasets, and to link astronomical data to astrophysical theory. Many branches of statistics are involved in astronomical analysis including nonparametrics, multivariate regression and multivariate classification, time series analysis, and especially Bayesian inference.
The registration to the 1st Italian Astrostatistics School is open. Up to 35 members; until Monday, priority is given to doctoral students. From Monday, if you are interested, all can register to the school.

What if we done the Schrodinger's cat experiment?

The Schroedinger's cat is a thought experiment proposed by Erwin Schroedinger in 1935 in order to proof the limits of quantum mechanics and Copenaghen interpretation. One of the most known consequence of the paradox is the many worlds interpretation, but also artist was inspired by the paradox. For example Chavdar Yordanov created a beautiful and funny animation with the music created by Sian Smith (via Open Culture).

Bioplastic from weaver's broom

At the end of 1980s the researcher Catia Bastioli developed a new bioplastic, Mater-Bi, using corn(1). Since 1989, there were developed a lot of bioplastic, for example the polymer poly(lactic acid)(2) or the broom fibers:
Fallico, C., Troisi, S., Molinari, A., and Rivera, M. F.: Characterization of broom fibers for PRB in the remediation of aquifers contaminated by heavy metals, Biogeosciences, 7, 2545-2556, doi:10.5194/bg-7-2545-2010, 2010
The present level of pollution, increasingly involving ground waters, constitutes a serious risk to the environment and also to human health. Therefore the remediation of saturated and unsaturated soils to remove pollutant materials is more and more frequently required. In the present paper, the possibility of removing heavy metals by permeable reactive barrier (PRB) from the groundwater carried out specifically with broom fibers, is investigated.
Once shown the economic benefits deriving from the use of this plant, a hydraulic characterization of the broom fiber mass was performed, determining the permeability and the porosity in correspondence to different levels of compactness of the fibers.
Having verified the effectiveness of removal of some heavy metals by these fibers, the results of some experiments, carried out in the laboratory for this purpose, are shown. These experiments were carried out utilizing broom fibers obtained in different ways and, limitedly to the considered pollutants, showed the high capability of these fibers to reduce their concentrations. The best results were obtained for the broom fibers extracted by a particular chemical-physical process.
Moreover, the behaviour of this fiber with time was investigated, determining the kinetic constant of degradation.
Gabriele, Bartolo, Teresa Cerchiara, Giuseppe Salerno, Giuseppe Chidichimo, Mabel Valeria Vetere, Cosimo Alampi, Maria Caterina Gallucci, Carmela Conidi, and Alfredo Cassano. "A new physical–chemical process for the efficient production of cellulose fibers from Spanish broom (Spartium junceum L.)." Bioresource technology 101, no. 2 (2010): 724-729. doi:10.1016/j.biortech.2009.08.014 (sci-hub)
A novel and efficient method for the extraction of cellulose fibers from Spanish broom (Spartium junceum L.) is presented. The method is based on the sequential combination between an initial chemical stage (alkaline digestion) and a subsequent physical–chemical stage, consisting of compression with hot air in an autoclave followed by rapid decompression (DiCoDe process, digestion–compression–decompression). The alkaline mother liquor deriving from the initial digestion step can be conveniently recycled after centrifugation followed by ultrafiltration. The process is characterized by the production of fibers with excellent physical–chemical properties, such as high mechanical resistance and high elasticity, and rapid production times. The fibers obtained after the DiCoDe process can be further softened and whitened by means of enzymatic digestion.
Fibers were morphologically characterized by scanning electron microscopy (SEM), while their composition and physical–chemical properties were determined by conventional methods, including colorimetry, TAPPI protocols, IR spectroscopy, and X-ray diffractometry.
Cassano, Roberta, Sonia Trombino, Ermelinda Bloise, Rita Muzzalupo, Francesca Iemma, Giuseppe Chidichimo, and Nevio Picci. "New broom fiber (Spartium junceum L.) derivatives: preparation and characterization." Journal of agricultural and food chemistry 55, no. 23 (2007): 9489-9495. doi:10.1021/jf071711k (sci-hub)
In the past decade interest in biopolymers has increased. Attempts were made to prepare new composite systems from biopolymers by binding different synthetic polymers to a biopolymer backbone. This paper reports the synthesis and characterization of derivatized broom fibers to prepare composites with either degradability or fireproofing properties. Synthetic strategies are described for the introduction of polymerizable functional groups or fluorine atoms on the glucose of cellulose chains of broom. The fibers containing polymerizable groups were copolymerized with dimethylacrylamide and styrene and, after that, investigated by optical polarizing microscopy (OPM) and scanning electron microscopy analysis (SEM). The materials containing fluorine were submitted to thermogravimetric analysis (TGA) and differential scanning calorimetry (DSC) for the purpose of verifying the fireproofing. Such derivatized biomaterials could be successfully used for applications in agriculture and in the packaging area.

(1) Bastioli, Catia. "Properties and applications of Mater-Bi starch-based materials." Polymer Degradation and Stability 59.1-3 (1998): 263-272. doi:10.1016/S0141-3910(97)00156-0 (sci-hub)
(2) Karamanlioglu, Mehlika, Richard Preziosi, and Geoffrey D. Robson. "Abiotic and biotic environmental degradation of the bioplastic polymer poly (lactic acid): A review." Polymer Degradation and Stability (2017). doi:10.1016/j.polymdegradstab.2017.01.009 (sci-hub)

The triple Klein bottle

Special thanks to @rudimathematici
Felix Klein was a German mathematician best known for a particularsurface thet he introduced in 18882: the Klein bottle, a non-orientable surface without edge, inside, outside and no boundary (for example a sphere is an orientable surface without boundaries).
In 1995 Alan Bennett, a retired glass-blower, became interested in Klein bottles and was in a unique position to satisfy his curiosity. From simple beginnings his researches produced a variety of beautiful and mathematically sophisticated forms. New discoveries have emerged from his work which formed the inspiration for this display.
This is one of a series of glass Klein bottles made by [the artist] for the Science Museum. It consists of three Klein bottles, one inside another. In the series Alan Bennett made Klein bottles analogous to Mobius strips with odd numbers of twists greater than one.
(via archive.org)

The aircraft of the future

Pierpaolo Lazzarini is an italian designer, who proposed an interesting concept personal aircraft for the future of mobility: I.F.O.

The project is an evolution of the previous U.F.O.

Pi and the Basel's problem

In 1644 the Italian mathematician Pietro Mengoli proposed the so-called Basel's problem, which asked for the exact solution to the square of the sum of the reciprocals of all the natural numbers: \[\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \cdots\] The solution to the problem came in 1735 thanks to Leonard Euler, at the time at the beginning of his brilliant career as a problem solver. The Swiss mathematician proved that the exact sum of the series is $\pi^2 / 6$.
The Euler's demonstration, published in its final form in 1741, is particularly interesting: Euler supposed that it's possible to apply the rules of the finite polynomials even those endless.
We start with the development in Taylor series for the sine function in 0: \[\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\] Dividing by $x$ both terms, we obtain: \[\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots\] whose roots are $\pi$, $-\pi$, $2\pi$, $-2\pi$, $3\pi$, $-3\pi$, $\ldots$ By changing the variable as $z = x^2$, the polynomial above becomes: \[\frac{\sin(\sqrt{z})}{\sqrt{z}} = 1 - \frac{z}{3!} + \frac{z^2}{5!} - \frac{z^3}{7!} + \cdots\] whose roots are $\pi^2$, $4\pi^2$, $9\pi^2$, $\ldots$
Now, given a polynomial $a_n x^n + \cdots + a_3 x^3 + a_2 x^2 + bx + 1$, for the formulas of Viète, we have that the sum of the reciprocals of its roots has as result $-b$. Applying this result for finished polynomials to infinite polynomial in $z$ above, we get: \[\frac{1}{3!} = \frac{1}{6} = \frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \frac{1}{16\pi^2} + \cdots\] and so: \[\frac{\pi^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots = \sum_{n=1}^\infty \frac{1}{n^2}\] It's simple to observe the connection between Mengoli's series and Riemann's zeta \[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}\] with $s=2$.
Last observation: in 1982 on the magazine Eureka, it appeared a rigorous proof of Euler's result signed by John Scholes, although it seems that such a demonstration circulated already to late sixties between Cambridge corridors.

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).

SphereX: the road to BB-8

BB-8 is the last Star Wars' droid introduced in The Force Awakens. It's a spherical robot with a free-moving head. Now, looking on arXiv, I found the proposal for SphereX, a new spherical robot for planetary explorations:
Wheeled planetary rovers such as the Mars Exploration Rovers (MERs) and Mars Science Laboratory (MSL) have provided unprecedented, detailed images of the Mars surface. However, these rovers are large and are of high-cost as they need to carry sophisticated instruments and science laboratories. We propose the development of low-cost planetary rovers that are the size and shape of cantaloupes and that can be deployed from a larger rover. The rover named SphereX is 2 kg in mass, is spherical, holonomic and contains a hopping mechanism to jump over rugged terrain. A small low-cost rover complements a larger rover, particularly to traverse rugged terrain or roll down a canyon, cliff or crater to obtain images and science data. While it may be a one-way journey for these small robots, they could be used tactically to obtain high-reward science data. The robot is equipped with a pair of stereo cameras to perform visual navigation and has room for a science payload. In this paper, we analyze the design and development of a laboratory prototype. The results show a promising pathway towards development of a field system.
The litle robot was tested under simulated lunar and martian gravity conditions, and the results are encouraging:
It was observed that as angle of separation between grouser decreases there is increase in average speed of robot and the power consumption remains almost constant. A hopping mechanism was developed for the robot that enables the robot to in theory perform unlimited hops. Currently the system is able to perform a hop of 8-10 cm under simulated Martian gravity. Extrapolating this, we would be able to achieve 16-20 cm hop in lunar conditions. The performance of hopping mechanism has to be improved to achieve the stated mission requirements. Based on power consumption for each hop and maximum power available, it was calculated that the robot would be able to produce maximum 208 hops in a single charge and robot would operate for 35 minutes of continuous hopping. The proposed SphereX design shows a promising pathway towards further maturation and testing of the technology in the field.

Life on Mars

This Voyager spacecraft was constructed by the United States of America. We are a community of 240 million human beings among the more than 4 billion who inhabit the planet Earth. We human beings are still divided into nation states, but these states are rapidly becoming a single global civilization.
We cast this message into the cosmos. It is likely to survive a billion years into our future, when our civilization is profoundly altered and the surface of the Earth may be vastly changed. Of the 200 billion stars in the Milky Way galaxy, some--perhaps many--may have inhabited planets and spacefaring civilizations. If one such civilization intercepts Voyager and can understand these recorded contents, here is our message:
This is a present from a small distant world, a token of our sounds, our science, our images, our music, our thoughts, and our feelings. We are attempting to survive our time so we may live into yours. We hope someday, having solved the problems we face, to join a community of galactic civilizations. This record represents our hope and our determination, and our good will in a vast and awesome universe.
Jimmy Carter


The IAU firmly opposes any discrimination based on factors such as ethnic origin, religion, citizenship, language, and political or other opinion and therefore expects U.S. officials to not discriminate on the basis of religion.
(read more)

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.