We cannot leave out time to begin with. Nor possibility, multiple times. A first attempt would be as follows. A system involves an input $I$ space, an state space $S$ and an output space $O$. We consider that the outputs are completely determined by states, that is, there is function $\phi: S \rightarrow O$. Let $T$ be time which has a total order $<$. Given a set $X$ let $X^T$ be the set of all maps $f : T \rightarrow X$. Then a system $\mathfrak{S}$ over $(I,S,O,\phi)$ is a function $\mathcal{S}: I^T \rightarrow \mathcal{P}S^T$.
$I$ has a distinguished element $\bot \in I$ which represents doing nothing. We write $f \triangle g$ for $f,g \in X^T$ if there is a $t \in T$ such that $ t' < t$ implies that $f(t') = g(t')$.
A link function is a function $\upsilon : O \rightarrow I$. With a link function we can define the plugging of one system into another. The link function can also be considered as $O \rightarrow I^n$ expressing simultaneous different outputs.
$I$ can be given the structure of a category so we can define composition, etc. We consider the possibility of multiple input and multiple output (and composition, plugging in of systems, including feedback). Thus there is an abstract cartesian product for a more general version of $I$ (and $O, \phi$, etc.).
We define $X^T_<$ to be the set of all functions $f : \downarrow t \rightarrow X$ where $\downarrow t =\{ t' \in T: t' < t\}$ for some $t$. Then another definition of system is a function $\mathcal{S}: I^T_< \rightarrow \mathcal{P}S^T_<$ which respects $t$.
We can consider a category $\mathcal{I}^T_<$ consisting of sets $I^{T^{t}_<}$ of functions $\downarrow t \rightarrow I$ for different values of $t$. If $t_1 < t_2$ then we have natural restriction map. This is a category. Likewise we can define categories $\mathcal{S}^T_<$ and $\mathcal{O}^T_<$. We can also define the category whose elements, for a given $t$, are sets of elements of $I^{T^{t}_<}$. A system is then a functor
\[ \mathcal{S}: \mathcal{I}^T_< \rightarrow P\mathcal{S}^T_<\]
$S$ is itself a moduli space of structures, networks. Thus $S = S_s \times S_d$ where the first component is a configuration and the second the global state.
Note that $I$ is more general than 'information', its can also include matter and energy as in biosystems.
In general our system will be a composition of other systems. We will need a calculus of dynamic reconfiguration , merging, separation, etc.
This second definition of system is better; it is the only one accessible to us, for we cannot observe infinite input,output or state histories.
Perhaps one of the most fundamental division is between systems having an origin (and perhaps an end) and those that are not postulated as having an origin. Thus we need a special state $\bot \in S$ of non-being. Then we can ask: does the system come into being because of a certain input (does this even make sense) or not ? We can conceive a kind of output from another system which is a self-replication or construction or generation of the new system. The state at the first instance $t_0$ in which the system has come to be, is its initial state $s_0 \in S$. Thus a generated system is given by a map (functor between categories) $\mathcal{S}: I^{T^{t_0}_<} \rightarrow \mathcal{P}S^{T^{t_0}_<}$ which respects $t$.
Note that the output or state-change is not necessarily to be considered instantaneous. Our framework is general enough to capture delay, feedback and a great variety of different kinds of causality, quasi-causality, indeterminism, etc. It is however important to be able to give the images $\mathcal{S}(s)$ in $\mathcal{P}S^{T^{t_0}_<}$ the structure of a $\sigma$-algebra and a measure (in particular complex-valued).
Thus a quantum system associates to each input path $p$ from an initial time $t_0$ to a time $t$ a space $\Omega_p$ of possible state paths from time $t_0$ to $t$ together with a (complex-valued) probability measure on $\Omega_p$. We assume that all the $\Omega_p$ are endowed with a $\sigma$-algebra structure in a coherence way (perhaps as induced by such a structure on the space of all state paths from $t_0$ to $t$).
If $T$ is the usual real line, then we can consider input sequences which are almost everywhere $\bot$ or only have a finite number of non-$\bot$ inputs. In this special case the framework above reduces to the classical Feynman formalization.
Again, this is a very interesting and inspiring post. In context of the application of category theory to quantum mechanics, I have identified several key points of interest and corresponding challenges. Here's my perspective on these insights.
ReplyDeleteCategory theory, with its emphasis on relationships over objects, provides a compositional language that naturally aligns with quantum phenomena such as entanglement and superposition. It offers an alternative framework to traditional mathematics, which may falter when grappling with the complex interactions within quantum systems. The holistic approach of category theory, via morphisms and objects, in my opinion can reflect the interconnectedness of quantum states and events better than traditional formalism.
When it comes to quantum causality, which blurs the lines between determinism and probabilism, category theory's flexible structure is well-equipped to accommodate the various causal frameworks that quantum theory presents (for example I, S and O in description of quantum states). This adaptability could contribute towards the unification of physics, a long-sought goal, by offering an abstract yet potentially universal language that could weave together the fabric of diverse physical theories.
However, the bridge between this high-level theoretical language and empirical data is not yet fully constructed. For category theory to contribute meaningfully to quantum mechanics, it must be interpreted in ways that yield testable predictions or offer new insights from quantum experiments. The abstraction that makes category theory so powerful is also its Achilles' heel, as it may obscure the path to empirical application and render the framework less accessible to those not versed in its complexities.
Furthermore, the development of physical intuition for category-theoretic concepts is a non-trivial task, one that poses a significant hurdle to its widespread adoption in physics. Nevertheless, the promise of category theory to inspire novel quantum algorithms and computational techniques for simulating quantum systems remains an exciting prospect.
On the other hand, however, even the commonly used formalism of quantum mechanics is considered unintuitive, and I think the important point here is to point out that our everyday intuitions are not necessarily correct. After all, neither quantum phenomena nor Einstein's relativity fall within their scope. Hence, I see this approach as a step in the right direction. The only question is how to combine it with physics, which unfortunately (I don't like to say this, as I feel I am primarily a theorist) needs experimental evidence. So what experimental advantages could category theory have in this context? Perhaps this requires a general change in methodology.
Nevertheless, as an abstract concept, this appeals to me far more than classical formalism, and I think a good start would be to try to capture particular constructs from quantum mechanics in this formalism.
I would probably start by trying to analyse operators in quantum mechanics and e.g. their self-adjointness, commutative relations, and extensions of group approaches, e.g. Lie groups, Poincare groups, conformal groups. The next question would be the modelling of measurement in this formalism.
Would pure states here be linked to the monoidal category? Could mixed states be described using quotient categories? Could forgetful functors be used when modelling irreversible transformations or even for modelling probabilities (input and output data)?
I have also been hit hard recently emotionally and somewhere out of my depth by an entirely different idea. A synthesis of category theory and fractal calculus in an attempt to answer questions about the EPR paradox, the black hole information paradox and the modelling of time in quantum mechanics, among others. Although I see this more as a first step towards something more general...
ReplyDeleteAgain, a picture is forming in my mind. On 23 March I have a seminar where I am talking about limitative theorems. It's possible that after that seminar I'll get some clarity on all this. At the moment I'm looking at various formalisms, category theory being my favourite. Nevertheless, it is very broad and abstract, and at this level of abstraction I am not able to formulate definitions in such a way that they are unambiguous to myself, let alone to others.
In one of your works you raised the issue of equality and identity. This is also my problem. Equality is one relationship, seemingly obvious, but I don't even know if it would find a place in this model.
I admit that I am sharing some thoughts rather loosely at the moment. Unfortunately English, although I am very familiar with it, is not a native language for me either, Polish is, and sometimes it seems to me that when I try to describe something similar in English, I sound artificial.
Of course, I am able to speak fluent English about simple things, about heavily fuzzy things that seem concrete. But if this is where the mathematical and linguistic subtleties come in, it is very difficult to express thoughts. The mathematics I would like to show has probably not been created yet, so when introducing anything I would also have to describe it in some language to explain what I mean by a symbol or by an operation that is new.
I have recently started learning Finnish. I don't have any grand plans to master it to perfection, but I notice many interesting aspects in linguistics that resemble problems in mathematics. Hence, I will be organising a conference on lexical openness. I am also writing a paper on lexical openness in theology with two co-authors. This paper is in English, so I will show it to you soon.
I think we'll write something together too, but I think there's even more value in this exchange and not having to rush or write to write. I really appreciate this blog. I have had quite little time lately. My apologies.
I see some really valuable things here and they really are unique.