Lawvere Thesis Statement

Maybe I should remind our physically inclined -Café readers that my questions cited by John above were part of an effort to understand what it means to say that

The syntax of quantum mechanics is quantum -calculus. #

This is a pretty cool statement!

In my attempts to decode this statement I first caught a mutant strain# of the true concept; and the -category doctor diagnosed conceptual sickness.

(I don’t know if it’s related, but over the week-end I also must have cought a more ordinary virus and became sick in the more traditional sense. Poor me.)

Given what John said here maybe I can phrase things like this:

A particular quantum theory is this:

1) a collection of Hilbert spaces , which are the spaces of -states.

For instance for the ordinary quantum mechanics of a single nonrelativistic particle there is just a single such Hilbert space, which is the space of states of precisely that particle.

As another example, in the quantum theory of a linearly extended thing like a string, we may have in general at least two different Hilbert spaces, of closed and of open string states.

Usually we want to admit several copies of the “single” quantum object we are describing, and hence we also consider all the tensor powers of the Hilbert spaces . For instance is the space of states of two closed strings.

It is a little easier to illustrate the following concepts in the string example than in the particle example, so I’ll stick to the string. If you don’t like strings you are invited to replace “string” by “1-dimensional spacetime” without loss of anything substantial here.

2) A quantum object in a given state may usually evolve into some other state. There are in general different “channels” along which to involve. Each such channel is characterized by the space of states of objects coming into the channel and the space of states of objects coming out of the channel on the other side.

So we are tempted to regard a “channel” (it’s really a common term in particle physics) as a morphism

(1)

For instance, a cylinder (usually called a cylinder diagram in this context) can be regarded as a channel

(2)

which describes a closed string coming in, just propagatinng without interaction and coming out again.

Slightly more interestingly, a sphere with three disks cut out (usually called the pair-of-pants diagram) may be regarded as a channel

(3)

In order to emphasize the relation of this business to the theory of (quantum) computation, we simply go ahead and rename everything encountered so far.

Instead of spaces of states we’ll talk about types. The elementary spaces of states (like and above) we call generating types.

Instead of channels we now say operations.

Composition of different channels usually obeys certain rules. For instance in the theory of “topological strings” we want a rule that the two ways to compose the pair-of-pants channel with itself lead to the same result. Such conditions we now call equations between functions.

Finally, we want to be able to consider different theories of topological strings, say, each of which involves the same sort of channels. In a lucky coincidence of terminology, different such theories are often called different models of the given quantum theory (like the “A-model” or the “B-model” of the topological string).

This “model”-terminology now becomes precise as follows.

We realize that all such theories can be regarded as functors

(4)

from 2-dimensional (open/closed) cobordisms to Hilbert spaces.

Whereas at the beginning I was thinking of quantum theories in all generality, right now I am making some sort of restriction to what are usually called “topological theories”.

As far as I understand, the crucuial point of this restriction for our current purpose is that the category is cartesian closed. For instance, the internal -object of morphisms from the circle to the circle is just

(5)

where the overbar denotes orientation reversal.

This now finally is getting close to answering the question what it means to say that quantum -calculus is the syntax of quantum mechanics.

Namely, is a CCC (a cartesian closed category), and, using the above language of types, functions and equations, a functor (preserving the relevant structure) from any to we would call a model of the -theory .

For the special case that we know that we may address such a functor equivalently as a model of topological “-particles (=-branes). For other CCCs we may still imagine addressing the functor as a quantum theory, though it might be an exotic sort of quantum theory that no physicist has ever dreamed of, I guess.

That’s currently roughly my understanding of what John and Mike are trying to tell me.

One thing I don’t understand yet is this:

is it reasonable to restrict the notion of “models in quantum mechanics” to functors whose domain is cartesian closed?

Doesn’t that exclude any “non-topological” theory?

For instance in ordinary non-relativistic QM of a single point particle we actually need 1-dimensional Riemannian cobordisms as domain.

Or the non-relativistic string. It needs 2-dimensional conformal cobordisms.

But these categories of cobordisms with extra structure are not cartesian closed, are they?

Is it therefore maybe better to say that “Quantum -calculus is the syntax of topological quantum mechanics”?

Tips and Examples for Writing Thesis Statements

Summary:

This resource provides tips for creating a thesis statement and examples of different types of thesis statements.

Contributors: Elyssa Tardiff, Allen Brizee
Last Edited: 2018-01-24 02:29:37

Tips for Writing Your Thesis Statement

1. Determine what kind of paper you are writing:

  • An analytical paper breaks down an issue or an idea into its component parts, evaluates the issue or idea, and presents this breakdown and evaluation to the audience.
  • An expository (explanatory) paper explains something to the audience.
  • An argumentative paper makes a claim about a topic and justifies this claim with specific evidence. The claim could be an opinion, a policy proposal, an evaluation, a cause-and-effect statement, or an interpretation. The goal of the argumentative paper is to convince the audience that the claim is true based on the evidence provided.

If you are writing a text that does not fall under these three categories (e.g., a narrative), a thesis statement somewhere in the first paragraph could still be helpful to your reader.

2. Your thesis statement should be specific—it should cover only what you will discuss in your paper and should be supported with specific evidence.

3. The thesis statement usually appears at the end of the first paragraph of a paper.

4. Your topic may change as you write, so you may need to revise your thesis statement to reflect exactly what you have discussed in the paper.

Thesis Statement Examples

Example of an analytical thesis statement:

An analysis of the college admission process reveals one challenge facing counselors: accepting students with high test scores or students with strong extracurricular backgrounds.

The paper that follows should:

  • Explain the analysis of the college admission process
  • Explain the challenge facing admissions counselors

Example of an expository (explanatory) thesis statement:

The life of the typical college student is characterized by time spent studying, attending class, and socializing with peers.

The paper that follows should:

  • Explain how students spend their time studying, attending class, and socializing with peers

Example of an argumentative thesis statement:

High school graduates should be required to take a year off to pursue community service projects before entering college in order to increase their maturity and global awareness.

The paper that follows should:

  • Present an argument and give evidence to support the claim that students should pursue community projects before entering college
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