and this one:

- Teiko Heinosaari, Mário Ziman,
*The Mathematical Language of Quantum Theory – From Uncertainty to Entanglement*, Cambridge University Press (2011) [doi:10.1017/CBO9781139031103]

added pointer to:

- Paul Busch, Marian Grabowski, Pekka J. Lahti,
*Operational Quantum Physics*, Lecture Notes in Physics Monographs**31**, Springer (1995) [doi:10.1007/978-3-540-49239-9]

I could not find…

Of course archive.org has the page

(here)

]]>Link

seem to be broken. I could not find a replacement link.

]]>added more textbook pointers:

Jun John Sakurai, Jim Napolitano, Cambridge University Press (1985, 2020) [doi:10.1017/9781108587280, Wikipedia]

Steven Weinberg,

*Lectures on Quantum Mechanics*, Cambridge University Press (2015) [doi:10.1017/CBO9781316276105]Chris Isham,

*Lectures on Quantum Theory – Mathematical and Structural Foundations*, World Scientific (1995) [doi:10.1142/p001, ark:/13960/t4xh7cs99]

adding textbook

- Peter Woit,
*Quantum Theory, Groups and Representations: An Introduction*, Springer 2017 [doi:10.1007/978-3-319-64612-1, ISBN:978-3-319-64610-7]

Quinn

]]>added pointer to:

- Robert Geroch,
*Geometrical Quantum Mechanics*, University of Chicago (1974) [pdf]

added pointer to:

- Ingemar Bengtsson, Karol Życzkowski,
*Geometry of Quantum States — An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [doi:10.1017/CBO9780511535048]

and will add this to a few related entries, too (such as to quantum state, quantum entanglement, …)

]]>added pointer to:

- Stéphane Attal,
*Quantum Mechanics*, Lecture 5 in:*Lectures on Quantum Noises*[pdf, webpage]

I have added pointer to:

- Karl Kraus,
*States, Effects, and Operations – Fundamental Notions of Quantum Theory*, Lecture Notes in Physics**190**Springer (1983) [doi:10.1007/3-540-12732-1]

While doing so, i have slightly re-arranged the list of textbooks under *References – General* in order to retain chronological and logical order.

added pointer to:

- Serge Haroche, Jean-Michel Raimond,
*Exploring the Quantum: Atoms, Cavities, and Photons*, Oxford University Press (2006) [doi:10.1093/acprof:oso/9780198509141.001.0001]

added pointer to:

- David Hilbert, John von Neumann, Lothar W. Nordheim,
*Über die Grundlagen der Quantenmechanik*, Math. Ann.**98**(1928) 1–30 [doi:10.1007/BF01451579]

I have added more publication data for:

John von Neumann,

*Mathematische Grundlagen der Quantenmechanik*(German) (1932, 1971) [doi:10.1007/978-3-642-96048-2]*Mathematical Foundations of Quantum Mechanics*Princeton University Press (1955) [doi:10.1515/9781400889921, Wikipedia entry]

added one more historical article:

- Max Born, Pascual Jordan,
*Zur Quantenmechanik*, Zeitschrift für Physik**34**(1925) 858–888 [doi:10.1007/BF01328531]

added pointer to some of the historical articles:

Werner Heisenberg,

*Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen*, Zeitschrift für Physik**33**(1925) 879–893 $[$doi:10.1007/BF01328377, Engl. pdf$]$Erwin Schrödinger,

*Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen*, Annalen der Physil**384**8 (1926) 734-756 $[$doi:10.1002/andp.19263840804$]$Paul A. M. Dirac,

*On the theory of quantum mechanics*, Proceedings of the Royal Society**112**762 (1926) $[$doi:10.1098/rspa.1926.0133$]$

added full publication data and hyperlinks to

- Klaas Landsman,
*Foundations of quantum theory – From classical concepts to Operator algebras*, Springer Open 2017 (doi:10.1007/978-3-319-51777-3, pdf)

I have added to quantum mechanics in the section “quantum systems” a subsection quantum subsystems with two basic definitions.

]]>Yes. I see your point.

Remark: The same prescription knowing at object level only of a Hilbert space (and not of underlying geoemtric space, what was my misconception of the translation in 0-dimensional case), this prescription keeping only Hilbert space data could also make any AQFT in any number of dimensions a 0+1-dimensional FQFT (this is in agreement that AQFT may be viewed a special case of QM). The fact that FQFT wants to keep (in another prescription like your AQFT vs FQFT article) the same dimension for traditional QFT is just a preference of the formalism to keep features of the underlying space which intentionally lost in QM case. I understand your point that you keep the dimension of evolutions multiple for QFT, in the FQFT convention. Thanks.

]]>It seems we have clarified this by private email now.

]]>It refers to **potentials**.

E.g. to the earlier extensive discussion on the difference between QM and QFT. I claimed that QFT is special case of QM and you that it is also true the opposite; and never got answered the main problem of your point of view about the inclusion of the space-dependent potential of a particular QM system. I can get the latter by a very unsatisfactory way, as mentioned above, to put all the possible space potentials on all possible Riemannian spaces simultaneously in the same category. This would mean a FQFT equivalent to all space potential QM systems at the same time. When you have TQFT then you distinguish different TQFTs, so one should also distinguish square well potential system from say Coulomb potential system in 3 dimension. The latter notions are not local notions but global in a sense. I understand that your point is to use the evolution operator as the part of a functor and every QM system has an evolution operator (in particular, every 4d AQFT also has an evolution operator and belongs to the same framework). But how do you distinguish different quantum mechanical systems. You have to take a different law of motion, different Hamiltonian, different evolution operator. But this operator is fed by the data in the underlying category. So if the underlying category is the category of Riemannian spaces, then I can take any polynomial scalar expression in terms of metrics and use it for Hamiltonian. Good, this is majority of examples in $n$Lab. Happy so far. But the potential is arbitrary, it does not take the metrics inside, it is just a function on the space. So the only way to take it is to put the function into the data. But I can hardly imagine a category of such, unless I put all possible potentials on all spaces in a category. But this is not the original QM system, this is the category of ALL QM systems of that type simultaneously. So the FQFT did not define a **particular** such system. FQFT defines all systems of the same structure type. Free fermionic theory for example. This is all possible free theories on all spaces. But Coulomb theory, well such a thing does not exist in FQFT world. Unless I misunderstand your point.

asked about very many times before and every time I get such a neglecting response.

Sorry, I don’t know what this refers to.

Don’t you see that this is one of the most important points of misunderstanding ??

I am not following. What is a point of misunderstanding?

]]>Thanks for your approach to time-dependence: this was as I mentioned less serious objection (I propose above another remedy), the more serious is the correspondence for arbitrary potentials. I was thinking about this for lonmg time, asked about very many times before and every time I get such a neglecting response. Don’t you see that this is one of the most important points of misunderstanding ??

]]>For time-dependent Hamiltonians (with whatever potential) one considers cobordisms with a map to a time-parameter space. Equivalently one can just mordify the cobordism category by having the objects be labeled in $\mathbb{R}$ and a cobordism of length $\ell$ go from object $t$ to object $t + \ell$.

]]>There is another thing which I see as a problem with the entry. The claim that one dimensional quantum mechanics is a Riemannian 0+1-dimensional field theory. So let me take an arbitrary example from textbooks of a space in k-dimensions $x_1,\ldots, x_n$ and a time-dependent potential $V(x_1,\ldots,x_n,t)$. Now the formalism of quantization described first has slight problem with time-dependent Hamiltonian (that is fixable by having more general form of the evolution operator) but the problem is that the potential is not something what the category of Riemannian manifolds knows of. It knows of differentiable manifold and a metric on it. So how do you put the arbitrary potential on a specific space into the data of a category ? This is nothing unusual, most exercises in QM courses will have such a specific potential. As long as one has the Hamiltonian depending only on Riemannian structure I have no problem with the approach. But I do not see how to put the Hamiltonians which depend on the data which is not in the category. Say, you have a Coulomb potential in 3 dimensions for a time-independent case. What do you do to the category of Riemannian manifolds to produce a functor whose data are equivalent to the QM of such a Coulomb system in 3-dimensions ?

Edit: it makes me very suspicious that it is possible to say, on the basis of what happens in one particular space with potential, what should functor do on all possible manifolds. Of course, one could consider ALL spaces with all potentials and put them somehow into the category, but this is cheating, as it does not distinguish different quantum mechanical systems (i.e.different potentials) as phrased in the usual way. I would say that a quantum field theory in 0+1-dimension is a general prescription on how to use the type of geometric data as an input for evolution operators (with worse case in the case of time-dependent potentials). This way diverging theories which do not make sense from the usual perspective and the converging theories which do are parts of the data of the same “field theory”. Of course if no potential than there are field theories whose all instances (that is for all input spaces) are equally OK from the usual formalism.

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