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in analogy to what I just did at classical mechanics, I have now added some basic but central content to quantum mechanics:
Quantum mechanical systems
States and observables
Spaces of states
Flows and time evolution
Still incomplete and rough. But I have to quit now.
I like the idea section but I am surprised in generality and poor structure required in the main part (definition etc.). I mean one usually takes the Hilbert space, and C-star algebra. Now having only star algebra, which may be in principle of uncountable infinity of dimensions makes me hard to believe that everything makes sense. GNS construction in uncountable number of dimensions ? I know that some people have ideas of working in different topoi, but this needs various justification and is certainly not helpful for the casual user wanting to see the non-disputed basics.
I think we should start with conventional case, where one has C-star algebra etc. and have special sections or entries for generalizations, which are many and in many direction.
I think it is good not to impose $C^*$-algebra structure in general. Many and most standard systems don’t come with bounded operators, and one has to bend over backwards to make make them such.
It is not easy to remove the C-star algebra. For example, what about he GNS construction I asked above ? Inner product in non-separable case ? How about KMS state and time evolution ? Some topological or categorical restrains are needed to make things contentful: some sort of spectral theory is needed to make sense of observables and measurement. Unbounded operators can be dealt in many ways, for example by looking at the associated multiplier algebra of the C-star algebra. One can also look at $W^*$-algebras. But, alternatively, dispensing with the spectral theory of Banach algebras or alike is really problematic. To talk about quantum mechanics it is not sufficient to have some abstract algebra, one has to make sense of measurement.
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.
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 $t + \ell$.
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 ??
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?
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.
It seems we have clarified this by private email now.
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.
I have added to quantum mechanics in the section “quantum systems” a subsection quantum subsystems with two basic definitions.
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