The n-Category Café

— From Schrödinger to Heisenberg picture —

– Schrödinger and Segal compared to Heisenberg and Haag-Kastler

In the so-called Schrödinger picture of quantum mechanics, state vectors are propagated in time.

In the so-called Heisenberg picture of quantum mechanics time-dependent observable operators are the objects under consideration.

The Schrödinger picture manifests itself in the conception of quantum field theory that goes back to Segal and says that:

An $n$-dimensional quantum field theory is a representation of a category of $n$cobordisms with extra structure..

This means that vector spaces “of states” are assigned to $(n-1)$-dimensional spaces, and that morphisms between these vector spaces are associated to the $n$-dimensional spaces cobounding these.

The refinement of this definition from representation 1-functors on $n$-dimensional cobordisms to $n$-functors on extended cobordisms is called extended (Segal-like) QFT. The definition of the charged $n$-particle that I am considering here is of this kind.

The Heisenberg picture of quantum mechanics, on the other hand, manifests itself as the other axiomatic formulation of quantum field theory that people have studied, known as algebraic quantum field theory (AQFT) or, sometimes, just local quantum field theory.

This definition says that

A (local, Lorentzian) $n$-dimensional quantum field theory on a space $M$ is something like a (co)sheaf of algebras (“of observables”) on $M$.

– The passage from Schrödinger to Heisenberg, 1-functorially –

I describe how the passage from the Schrödinger to the Heisenberg picture looks like in more or less abstract, arrow-theoretic terms. I do this explicitly for the $(n=1)$-particle only, but the idea is that the way this is formulated the passage to higher $n$ is already visible.

(The following paragraph has grown out of this comment.)

- from states to observables, for $n=1$ -

Here is a worldline $$\mathbb{R}$$ and here is the category of paths on the worldline: $$P_1(\mathbb{R}) \,.$$ A quantum mechanical system gives rise to a propagation functor $$U : P_1(\mathbb{R}) \to \mathrm{Hilb} \,.$$ In other words, this is a bundle of Hilbert spaces over $\mathbb{R}$ with connection $$H \,dt \,,$$ where $H$ is the Hamiltonian. (Compare the discussion with Bruce.)

We may want to refine this a little: the Hilbert spaces that we assign to each point in time are special, in that they appear as spaces of sections of the charged particle. In particular, this means, in general, that these are pointed Hilbert spaces, in which a particular cyclic vector $$v = |0\rangle$$ is singled out, usually called the ground state or the vacuum.

It so happens that there is an alternative way to think about Hilbert spaces with chosen vacuum vectors: namely in terms of $C^*$-algebras with with a chosen state on them.

From every Hilbert space $H$ with cyclic vector $v \in H$, we get the $C^*$-algebra $B(A)$ of bounded operators on $H$, equipped with the pure normal state defined by $a \mapsto \langle v, a v\range$.

This should in fact be a functor $$B : \mathrm{Hilb}_{\mathrm{cyc}} \to C^*_{mathrm{sts}} \,.$$

Conversely, given any $C^*$-algebra $A$ with a pure normal state $\phi$ on it, the GNS construction provides a Hilbert space $H$ with a cyclic vector $v \in H$ such that $A \simeq B(H)$ and $\phi(a) = \langle v, a v\range$ . This should give a functor $$\mathrm{GNS} : C^*_{\mathrm{sts}} \to \mathrm{Hilb}_{\mathrm{cyc}} \,.$$ Moreover, these two functors should yield an equivalence of categories. (I am very grateful to Masoud Khalkhali for his help with figuring out what the precise formulation of this statement is.)

Given this (presumable) equivalence of the codomain of the Schrödinger picture functors $U : P_1(X) \to \mathrm{Hilb}_{\mathrm{cyc}}$ with the category $C^*_{\mathrm{sts}}$, there is nothing more natural than passing to that equivalent picture by hitting $U$ with the functor $$B_* : [P_1(\mathbb{R}),\mathrm{Hilb}] \to [P_1(\mathbb{R}),C^*] \,,$$ which postcomposes any functor with the functor $B : \mathrm{Hilb}_{\mathrm{cyc}} \to C^*_{\mathrm{sts}}$.

The resulting Heisenberg transport $$B_* U : P_1(\mathbb{R}) \to C^*$$ is now an algebra bundle with connection over $\mathbb{R}$!

(Each fiber should be thought of as the closure of the Weyl algebra characterizing the phase space of the system. )

The connection 1-form now reads $$\mathrm{ad}_H \, d t \,.$$

To be explicit, assume the QM system is that of a particle propagating on $\mathbb{R}^n$ (I could consider $\mathbb{R}^1$ for simplicity, but I want to keep us from confusing target space with parameter space).

Then, all the Hilbert space fibers of the original bundle are canonically identified with $L^2(\mathbb{R}^n)$.

Accordingly, all the algebra fibers are canonically identified with the Weyl algebra, which I think of, for convenience, as canonically generated infinitesimally from elements $x^i$ and $p_i$ with $[x^i,p_j] = i \delta^i_j$.

- sheaves of algebras of observables for $n=1$ -

Now consider the algebra of flat sections of our algebra bundle over the worldline.

Better yet, consider the sheaf of algebras of flat sections of this bundle.

To every open interval $(a,b) \subset \mathbb{R}$, this will associate all flat section of $B_* U|_{(a,b)} : P_1(a,b) \to C^*$.

But every such flat section is completely determined already by its value over any one codimension one space – i.e. by its value of any one point.

Assume the points $0$ and $t$ are inside our interval.

Over $0$, we have the algebra element $x^i$. We might call that $$x^i(0)$$ to make explicit that we think of $x^i$ as sitting in the fiber $B_* U(0)$ and not anywhere else.

This element determines an entire flat section over all of $(a,b)$. Over $t$, the value of this flat section will be (for the free particle) $x^i - \frac{t}{m}p_i$, now regarded as an element of $B_* U(t)$. So we should, following the above convention, write this as $$x^i(t) - \frac{t}{m}p_i(t) \,.$$

Let me write $$e_{x^i(0)} \in \Gamma(B_* U)$$ for the flat section uniquely specified by that fact that its value at $0$ is $x^i$.

Similarly I then write $e_{x^i(t) - \frac{t}{m}p_i(t)}$ for the flat section uniquely specified by having the value $x^i + \frac{t}{m}p_i$ over $t$.

Since both $x^i(0)$ and $x^i(t) + \frac{t}{m}p_i(t)$ uniquely specify the same flat section of our algebra bundle, we have $$e_{x^i(0)} = e_{x^i(t) + \frac{t}{m}p_i(t)} \,.$$ If we were in a more relaxed mood, we might simply write $$x^i(0) = x^i(t) - \frac{t}{m}p_i(t) \,.$$

(I know I require a certain tolerance for sophisticated-looking trivialities here. The point is that this exercise is supposed to make life easier as we go up the dimensional ladder.)

Now, sections of a bundle of algebras form an algebra themselves, simply by pointwise multiplication.

For instance, we find $$[e_{x^i(0)}, e_{p_j(0)}] = e_{i\delta^i_j\mathrm{Id}(0)} \,.$$ Or $$[e_{x^i(0)}, e_{x^j(t)}] = [e_{x^i(0)}, e_{x^j(0) + \frac{t}{m}p_j(0)}] = \frac{t}{m} e_{i\delta^i_j \mathrm{Id}(0)} \,.$$

In our more relaxed mood, this reproduces the familiar formulas for the operator algebra of fields on the worldline $$[x^i(0), x^j(t)] = \frac{t}{m} i\delta^i_j \,.$$

Bottom line: (sheaves of) operator algebras of fields are (sheaves of) algebras of flat sections of the Heisenberg bundle obtained as the push-forward of the Hilbert bundle obtained in the Schrödinger picture.

- cosheaves and Haag-Kastler nets of algebras of observables for $n=1$ -

A Haag-Kastler net of observables is defined to be something rather similar to a (pre)cosheaf of algebras. Is that at odds with the sheaves of algebras of observables that we found above (and that are known in Euclidean field theory)?

The sheaves of observables that we have found here are actually rather special. The way they arise, we see that we may just as usefully think of them as cosheaves:

since every flat section is uniquely determined by its value on a codimension one subset (a point in our $n=1$ example), we have a canonical injection of the algebra $A_V$ of flat sections over $V$ into the algebra $A_U$ of flat sections over $U$ for open sets $$V \subset U \,.$$

This is really crucially due to the fact that we are dealing with flat sections of the Heisenberg algebra bundle.

Accordingly, it makes good sense to think of our flat sections as forming a cosheaf of algebras.

Or, alternatively, we may think of them as forming a precosheaf of algebras with the special condition that the corestriction maps are injections.

The latter definition is that appearing in the definition of Haag-Kastler nets of observables.

The full definition of Haag-Kastler nets refers to a Lorentzian structure on parameter space. The presence of such a structure is something I don’t want to restrict attention to. I would like to understand Euclidean and Lorentzian parameter spaces for QFTs (of worldvolume theories of charged particles) on a commen basis.

For that reason, I will talk about Haag-Kastler-like nets of algebras, in the following, whenever I am referring to pre-cosheaves of algebras with injective corestriction maps.

It turns out (at least as far as I can see) that this concept (rather than that of a sheaf of algebras) is the one we need to really perform the next step.

— From Heisenberg to Schrödinger: the Worldvolume Anomaly —

In the practice of quantum field theory, people often find themselves faced with the problem opposite to the one discussed so far: on local patches of parameter space algebras of observables are known, and one tries to reconstruct from that a Hamiltonian that generates the time evolution, globally.

– finding a Hamiltonian –

Suppose on parameter space (in the example that I am concentrating on, this is the real line, $\mathbb{R}$, or rather its categorical incarnation, $\mathrm{tar} = P_1(\mathbb{R})$) we have given a Haag-Kastler-like net of algebras (of observables), i.e. a pre-cosheaf of algebras with injective corestriction maps.

On each patch $U$ of target space, a local candidate for a Hamiltonian is an element $$H_U \in A_U$$ of the algebra associated with $U$, such that, for $V \subset U$ a subset and $V_t \subset U$ its translation by $t$ (which makes sense in my example, where $n =1$, and has an obvious generalizations to $n \gt 1$) such that $$\mathrm{Ad}_{e^{i t H_U}} : A_V \simeq A_{V_t} \,,$$ where this is interpreted in $A_U$ using the injective corestriction maps.

In words: an element of the algebra is a candidate for a local Hamiltonian if its action on the algebra can be regarded as the relevant (time-)translation operation.

– gluing the Hamiltonian –

From our discussion of the passage from the Schrödinger to the Heisenberg picture, we know that the Hamiltonian is actually (the single component of) a globally defined connection 1-form on the Hilbert bundle of states, or, equivalently (by its adjoint action) on the Heisenberg bundle of algebras of observables.

In order to reconstruct this situation from our local data we need to

- cover parameter space by open sets

- glue the algebras of observables to sections of an algebra bundle that are flat with respect to the local Hamiltonians

- such that these local Hamiltonians constitute a globally defined connection 1-form on this bundle of algebras.

If this last step fails, i.e. if we have a locally (local on parameter space!) well-defined Hamiltonian which fails to constitute a globally well defined Hamiltonian, we say we have a worldvolume anomaly.

(For the case of a 2-particle being a string with a conformal field theory on its worldvolume (a worldsheet, in this case), this anomaly is precisely (a not so common incarnation of) the famous conformal anomaly).

It is helpful to consider this problem that we are faced with here in the general context of bundles with connection.

For $U$ any patch, let $P_1(U)$ be the category of paths in that. A vector bundle with connection on $U$ is a parallel transport functor $$\mathrm{tra}_U : P_1(U) \to \mathrm{Vect} \,.$$ A flat (covariantly constant) section $e$ of this bundle is a morphism of functors $$e_U: 1 \to \mathrm{tra}_U \,,$$ where $1$ denotes the tensor unit in the monoidal category of all such functors.

In components this means that for every path $x \stackrel{\gamma}{\to} y$ in $U$ we have $$\array{ \mathbb{C} &\stackrel{e_U(x)}{\to}& V \\ \mathrm{Id}\downarrow \;\; && \;\; \downarrow \mathrm{tra}_U(\gamma) \\ \mathbb{C} &\stackrel{e_U(y)}{\to}& V } \,.$$ On the intersection of $U$ with another open set $V$, with transition morphism $$g_{U,V} : \mathrm{tra}_U \to \mathrm{tra}_V$$ we find the situation $$\array{ \mathbb{C} &\stackrel{e_U(x)}{\to}& V &\stackrel{g_{U,V}(x)}{\to}& V \\ \mathrm{Id}\downarrow \;\; && \;\; \downarrow \mathrm{tra}_U(\gamma) && \;\; \downarrow \mathrm{tra}_V(\gamma) \\ \mathbb{C} &\stackrel{e_U(y)}{\to}& V &\stackrel{g_{U,V}(y)}{\to}& V } \;\; = \;\; \array{ \mathbb{C} &\stackrel{e_V(x)}{\to}& V \\ \mathrm{Id}\downarrow \;\; && \;\; \downarrow \mathrm{tra}_V(\gamma) \\ \mathbb{C} &\stackrel{e_V(y)}{\to}& V } \,.$$ On top of this ordinary gluing law, we will now have to require that, on $U \cap V$, the two parallel transport functors actually coincide: we want to find a globally defined parallel transport functor on a trivial bundle, which is then to be interpreted as our Schrödinger picture time evolution operator.

This means, in our case, in particular, that the transitions with which we glue the Heisenberg algebra bundles over double intersections have to be operators that commute with the local Hamiltonians.

In the language of quantum field theories, such operators which commute with the Hamiltonian are addressed as symmetries of the system. Hence we find that,

In order to find a non-anomalous worldvolume theory, we need to be able to consistenly glue the local algebras of observables by symmetries of their local Hamiltonians.

– a very simple example –

(in the next issue – enough for today)