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have created a stub for supersymmetric quantum mechanics
Zoran, I see that you once dropped a big query box at quantum mechanics with a complaint. I disagree with the point you make there: we have fundamental definitions of quantum field theory and restricting them to 1 dimension gives quantum mechanics. If you want to turn this around and understand all QFTs as infinite-dimensional quantum mechanics (which, yes, one can do) you are discarding the nice conceptual models and kill the concept of extended QFT.
In any case, I think remarks like this (in the style of “we can also regard this the other way round like this”) are better added into an entry as what they are – remarks – than as query boxes that give the impression that there is something fishy about the rest of the entry.
Urs, quantum mechanics (link above is incorrect) knows in general nothing about space, numbers of degrees of freedom or alike. The space etc. appears once you write down some representation like x-representation or so. This formalism, called quantum mechanics from early 20th century is fully general: Hilbert space, states, Hamiltonian as (up to “i”) the Hermitean generator of the evolution. Classical example: spin up and spin down, and the Feynman’s lectures derivation of quantum mechanics just from principle of unitarity what gives you that the generator of evolution is a Hermitean operator. Such basic of QM, never see “space” and “number of degrees of freedom”. They appear in some special cases of quantum mechanics, including QFT, QM of a finite Hilbert space, QM of finitely many degrees of freedom and so on. Neither of these special cases is fully general (for example it is not known if the string theory at quantum level, if ever fully explained can be shown equivalent to a QFT. maybe yes, maybe not, but whatever it is it believed that it will be a quantum mechanics: Hilbert space, Hamiltonian and probability interpretation). Also “QFT on the lattice” is not QFT in its standard sense in nLab. But it is just a special case of QM again: Hilbert space and so on (and this is the way most physicists teach, including Feynman, so why we do n ot do it this way in nLab). This whole thing is just terminological. (edit: Classical mechanics of particles and fields. Quantum mechanics of particles and fields. These are standard expressions and even book titles.)
Now besides this terminological discussions (I had taken about 20 courses in quantum mechanics and QFT for physicists, and read for few years through most standard textbooks, Sakurai, advanced Sakurai, Schiff, Feynman, Landau, Blohincev, Sudbery, Bjorken-Drell, Scadron, Bogoliubov, Akhiezer etc. and have some feeling of the tradition here), I am less categorical, but somewhat suspicious (that part you will probably answer, cooling my suspicion to zero, but still I think it is better to stick toward the formalism/terminology of founders of QM, like von Neumann who defined what QM is in full generality (Hilbert space etc.), where QFT fits as an example and which is earlier than the discovery of Dirac see and other corrections to naive systems with infinitely many degrees of freedom, leading to modern QFT) about the claim that if I forget the convention and just take the content that all finite-dimensional QM systems in narrow sense (that is just QM of finitely many bosonic and fermionic degrees of freedom) are obtainable from FQFT. Thus let X be the Euclidean plane with an “external” potential U(r,t) depending on position and time. Consider just a single particle in that time-dependent and position-dependent potential. What is the corresponding functor (and what is the base “cobordism” category, which takes care of time-dependent potentials) ?
Edit: the only thing I see is tautological, and could be applied to QFT as well, and having just specified time intervals as the geometries, even for infinitely many degrees of freedom. On the other hand, it would be interesting to see cobordism approach for noncommutative QFTs which are now popular among my colleagues in Zagreb and their collaborators in India. Still they are just examples of QM.
Urs, why not taking definition of QM as in von Neumann “Foundations”, with Hilbert space, Hamiltonian and so on, as basic.
Then one can have another, modern section on a relation between QM of finitely many degrees of freedom and FQFT approach. This would agree both with the traditional terminology and your wish to put into nPOV.
BTW, I did not read the famous Dirac’s classic book on QM. I could not afford the book at the time, and when I could I found it boring in view of knowing other references already.
I removed the query in quantum mechanics discussing with Ian about quantum physics vs. quantum mechanics.
Zoran: I slightly disagree with that opposite extremity either. The sentence is better suited to define quantum physics (though complement to classical physics includes relativity, while opposite to classical mechanics includes thermodynamics). Quantum physics is not the same as quantum mechanics. There are quantum phenomena which are not treated by quantum mechanics only. I mean it would be very unusual calling quantum statistical physics, part of quantum mechanics; it requires special limiting assumptions (thermodynamic limit, quantum ergodicity and so on lie outside) outside of scope of the things derivable from quantum mechanics.
Ian Durham: J.J. Sakurai’s book Modern Quantum Mechanics (not to be confused with his Advanced Quantum Mechanics which is a field theory book) discusses quantum statistical mechanics under the guise of quantum mechanics as does the forthcoming book Q-PSI: Quantum processes, systems, and information by Schumacher and Westmoreland. While it may not be entirely accurate, “quantum physics” and “quantum mechanics” are usually treated as being synonymous.
Zoran: Quantum physics includes all quantum phenomena, including experimental reality. Quantum mechanics is just the theoretical description at the fundamental level, like mechanics is of the classical physics.
Ian Durham: I agree, in principle, but I’m simply saying that not everyone interprets the term as such.
Zoran: not everybody cares about fine distinctions, but we should try to acknowledge the distinctions which are made at least by more careful treatises, not the least common denominator.
Ian: The more I think about it, the more I strongly disagree. I think you are correct in saying that quantum mechanics is the theoretical formalism for describing quantum phenomena in a non-field theoretic POV. But what we’re going on this site is describing the theory in terms of categories, not the phenomena. If you want to strip it all down to the bear bones, rid quantum physics of the “mechanics,” and then rebuild it entirely from an nPOV, be my guest. But so far, that is not what has been done on this site.
Zoran: why non-field ?? Field is a continuous limiting case of mechanics when it becomes with infinitely many degrees of freedom. Classical mechanics of particles and fields. Quantum mechanics of particles and fields. These are standard expressions. After 52 courses of physics I attended in my life I am confident in basic terminology. Quantum physics is without a doubt more general notion. For example, Bohr was studying quantum physics, and de Broglie was and they did NOT do QM as they studied quantum physics at a incomplete level before QM proper was invented. Sometimes one intermediate phase, from 1913 with Bohr-Sommerfeld quantization conditions call “old quantum mechanics”, though. The things you talk about categories and $n$POV have nothing to do with all of this and make no sense in this discussion.
So here is the other query, the one Urs was complaining about
Zoran: While this has some merit for this particular direction/subfield, it is other way around: quantum field theory is a special kind of a quantum mechanical system (with infinitely many degrees of freedom). You are here taking very special point of view and identifying quantum mechanics with quantum mechanics of a system with finitely many degrees of freedom. One teaches thus other way around: there are quantum mechanical systems, then there are some special kinds including QFTs. Quantum mechanics of the point particles on the other hand is not subsumed by the formalism below either. For example, the time-dependent Hamiltonians and quantum mechanics for finite state systems do not seem to be included. I personally do not understand how do you put a concrete potential in the game below either but I suppose it is a way. I mean which space-dependent and other parameters are allowed for $H$ and how it works (note that while the usual forces are related to the curvature, I do not see there is a mechanism for all kind of potentials and more general hamiltonians to be just derived from general rule and be invariant under isomorphism in Riemannian category) ?
Toby: It looks like Urs understands ’mechanics’ in much the same way that I did at classical mechanics. I do agree that that the term includes time-dependent Hamiltonians and similar generalisiations, however.
I reordered the entry and included both points of view simultaneously. Waiting for the time-depending explanation to be added by Urs at some point. (Edit: I have no problem in writing the time-ordered unitary evolution from time dependent Hamiltonian, but with the dependence of the objects of the cobordism category of the time-feature: on the other hand the point of view of hiding the space-time into the Hilbert space could be used to look at n-dimensional QFT as n-k-dimensional QFT, what does not sound right for classification purposes, is it essential?)
We should also have more on Heisenberg vs. Schroedinger, interaction and other pictures as well.
quantum mechanics (link above is incorrect) knows in general nothing about space, numbers of degrees of freedom or alike.
I think the space you mean is parameter space. Quantum mechanics has evolution only along one single parameter. Hence it is 1-dimensional QFT.
I meant the configuration space. I understand what you are saying. That does not mean that AQFT is not a special case of von Neumann general formalism of quantum mechanics from late 1920s.
FQFT takes space-time defineable in a class of systems into a special role.
Question: As in mirror symmetry we have different categories of manifolds in consideration of symplectic and of complex side, I suppose we have different cobordism categories. In which sense the two are equivalent as FQFT ? It is clear to me in A-infinity and in Hilbert space formulation as SCFTs, but what about the FQFT point of view? I mean what is the equivalence of FQFT if we allow change of domain of functors ?
Just to prevent misunderstanding: “link above is incorrect” meant that in @1 you gave a misspelled link name (so I quoted it again). Just in the case that you answered on that. Edit: I now see you corrected it, so it was not that, good.
P.S. is nLab still working ? Can not access quantum mechanics.
I have expanded the entry supersymmetric quantum mechanics a little, to go along with this Physics.SE comment
I have cross-linked supersymmetric quantum mechanics (here) with Witten genus a bit more (here) trying to make more explicit that it is the would-be Dirac-operator-on-smooth-loop-space-realization of the Dirac-Ramond operator which would rigorously realize superstring quantum dynamics as supersymmetric quantum mechanics on loop space (this is how the Morse theory business of susy QM was found by Witten originally).
added pointer to discussion of supersymmetric quantum mechanics in the perspective of supergeometry (integration over supermanifolds, picture changing operators):
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