COURSE NOTES ON QUANTIZATION AND COHOMOLOGY, WINTER 2007

JOHN BAEZ* NOTES BY APOORVA KHARE*

Contents

1. Jan 16, 2007: Schrödinger’s equation 3 1.1. Questions 3 1.2. Motivating geometric quantization 5

2. Jan 23, 2007: Categorification 7 2.1. A secret functor 7 2.2. Bringing in arbitrary categories 8

3. Jan 30, 2007: Physics is rigged! 10 3.1. The analogous viewpoints 10 3.2. Switching between the classical and quantum viewpoints 10 3.3. Wick rotation and a spring in imaginary time (revisited) 12

4. Jan 23, 2007: Statistical mechanics and deformation of rigs 13 4.1. Statistical mechanics “quantizes” strings 13 4.2. A family of rigs via the Boltzmann map 14 4.3. The analogous situation for quantization 15

5. Feb 13, 2007: An example of path integral quantization - I 16 5.1. Example: free particle on the real line 16 5.2. Doing the math 17

6. Feb 20, 2007: An example of path integral quantization - II 20 6.1. Time-evolution operators 20 6.2. Bringing in the Hamiltonian 21 6.3. Computing the normalizing factors 22

7. Feb 27, 2007: More examples of path integrals 25 7.1. A potential problem 25 7.2. The Lie-Trotter Theorem and self-adjoint operators 26 7.3. Generalization to complete Riemannian manifolds 27 7.4. Back to the general picture 27 7.5. Digression of the day: Cauchy surfaces 28

0^{∗}Department of Mathematics, University of California at Riverside, USA

8. Mar 6, 2007: Hilbert spaces and operator algebras from categories 29 8.1. (Pre-)Hilbert spaces from categories 29 8.2. Operators and multiplying them 32

9. Mar 13, 2007: The big picture 33 9.1. The case of finite categories 33 9.2. Example: particle on a line 33 9.3. From particles to strings 34 9.4. Digression on torsors 35

1. Jan 16, 2007: Schrödinger’s equation

These are lecture notes taken at UC Riverside, in the Tuesday lectures of Professor John Baez’s Quantum Gravity Seminar, Winter 2007.

Classical (p = 1)

Quantum (p ≥ 0)

(1) Lagrangian Mechanics. The path between two points (t_{i}, q_{i}) is γ ∈ P_{q_{0}→q_{1}} = P_{q_{0}→q_{1}}Q satisfying δS(γ) = 0. Here,

• Q is the configuration space,

• P_{q_{0}→q_{1}} is the path space,

• {γ : [t_{0}, t_{1}] → Q : γ(t_{i}) = q_{i}},

• L : T Q → ℝ is the Lagrangian, and

• S(γ) := ∫_{t_{0}}^{t_{1}} L(γ(t), γ̇(t)) dt is the action.

(1) Lagrangian Mechanics. The position of a particle at time t is given by a wavefunction ψ_{t} : Q → ℝ. Moreover, the amplitudes |ψ_{t}| form probability densities, so ψ_{t} ∈ L^{2}(Q) for all t. To consider “paths”, we now have to integrate over all initial points and all paths ending at q_{1}. Thus, ψ_{t_{1}}(q_{1}) now equals

∫_{q_{0}∈Q} ∫_{P_{q_{0}→q_{1}}} e^{iS(γ)/ℏ}ψ_{t_{0}}(q_{0})(𝒟γ) dq_{0}

[figure: Q at both ends + path γ ∈ P_{q_{0}→q_{1}}] [figure: Q at both ends + endpoint wavefunctions ψ_{t_{i}} + lots of paths]

(2) Hamiltonian Mechanics. We now have the phase space X = T^{∗}Q (or any symplectic manifold), with H : X → ℝ the Hamiltonian function. (In local coordinates (p_{i}, q^{i}), and using the Legendre transform λ : T Q → T^{∗}Q, we have H = p_{i}q̇^{i} − L ∘ λ.) Then γ̃(t) := (q(t), p(t)) ∈ X satisfies Hamilton’s equations:

\frac{d}{dt}γ̃(t) = v_{H}(γ̃(t))

where v_{H} is the Hamiltonian vector field, with dH = ω(v_{H}, −), where ω = −dα is the symplectic structure.

(2) Hamiltonian Mechanics. Each ψ_{t} ∈ L^{2}(Q) satisfies Schrödinger’s equation:

\frac{d}{dt}ψ_{t} = \frac{1}{iℏ}Ĥ(ψ_{t})

(look at it as ψ : ℝ → L^{2}(Q), say,) where Ĥ : L^{2}(Q) → L^{2}(Q) is a linear operator obtained (somehow!) by “quantizing” H.

1.1. Questions. This chart raises lots of questions. (We mention a couple of them here.)

(1) How do you do “path-integrals” ∫_{P_{q_{0}→q_{1}}} 𝒟γ over the path space?

Apparently, there is no meaning to the “measure” 𝒟γ (or it has not yet been found!), but there is, to e^{iS(γ)/ℏ}𝒟γ, at least in wellbehaved cases. One such has been extensively studied by physicists: given a smooth finite-dimensional manifold Q, define

L(q, q̇) = \frac{m}{2}||q̇||^{2} − V (q)

where we have the obvious kinetic and potential components. (If we replace i by −1 in the case of the harmonic oscillator V (q) = |q|^{2}, then the expression above, namely, e^{−S(γ)/ℏ}𝒟γ is the Wiener measure.)

Digression on complete Riemannian manifolds: Note that to define the above Lagrangian, we need additional assumptions on Q, V. Namely, the kinetic component needs a metric, so we assume that the manifold is Riemannian. Moreover, we want “closed” manifolds, so that particles “don’t fall off the edge” - so we assume that Q is complete as a metric space. Here, we have

Definition 1.1. Given a connected Riemannian manifold M,

(a) M is a metric space if we set the distance between points m_{0}, m_{1} ∈ M to be the Riemannian distance:

d(m_{0}, m_{1}) := inf_{γ∈P_{m_{0}→m_{1}}M} |γ|

where P_{m_{0}→m_{1}}M is the set of all piecewise regular curves (i.e. γ̇(t) is zero or undefined at most at finitely many points) from m_{0} to m_{1}, and given any (smooth) parametrization γ : [t_{0}, t_{1}] → M, we have its (parametrization-independent) length

|γ| := ∫_{t_{0}}^{t_{1}} |γ̇(t)| dt

Moreover, the metric topology is the same as the manifold topology.

(b) M is geodesically complete if every maximal geodesic is defined for all t ∈ ℝ.

Then (for “completeness’ sake” ¨⌣!) we have the following result, that tells us a consequence of completeness:

Theorem 1.2 (Hopf-Rinow Theorem). A connected Riemannian manifold is complete (as a metric space) if and only if it is geodesically complete.

Moreover, this is if and only if any two points can be joined by a geodesic - which is why this is relevant to us.

Back to our discussion: We thus define L(q, q̇) as above, using the assumption that Q is a connected complete Riemannian manifold (to keep our particle from “falling off the edge”), and V : Q → ℝ should be smooth and bounded below (again to keep our particle from acquiring a lot of kinetic energy, and “shooting off to infinity” in finite time).

References. For the basic ideas, try Feynman and Hibbs, Quantum Mechanics and Path Integrals.

For mathematical rigor, try Barry Simons’ Functional Integration and Quantum Physics.

(2) How do we get the Hamiltonian operator Ĥ : L^{2}(Q) → L^{2}(Q) from

the Hamiltonian function H : T^{∗}Q → ℝ?

In some cases, it is easy to write down Ĥ, e.g. under the same assumptions we made while discussing the previous question:

H(q, p) = \frac{|p|^{2}}{2m} + V (q)

(where H is a connected complete (finite-dimensional) Riemannian manifold, and V is smooth and bounded below). In this situation, Schrödinger wrote:

Ĥ = −\frac{ℏ^{2}}{2m}∇^{2} + mult_{V}

where ∇^{2} := ∇^{⟶} · ∇^{⟶} is the Laplacian (and ∇^{⟶} the gradient). This sends a square-integrable function f on Q to

f ↦ −\frac{ℏ^{2}}{2m}( ∇^{⟶}f, ∇^{⟶}f) + f · V

(where we compute the first term using the Riemannian metric). This is often written simply as −\frac{ℏ^{2}}{2m}∇^{2} + V. Schrödinger got this by guessing the quantization rule: p ↦ \frac{ℏ}{i} ∇. (There were physical motivations for such a guess - namely, results for waves.) This yields:

\frac{|p|^{2}}{2m} = \frac{(p, p)}{2m} ↦ −\frac{ℏ^{2}}{2m}∇^{2}

1.2. Motivating geometric quantization. Ideally, we would like • a method to get Ĥ from more general H.

• to use our assumptions (on Q and V) to show “good” properties of

Ĥ. For instance, if A : K → K is a self-adjoint operator on a Hilbert space K,

then e^{iAt} : K → K is well-defined and unitary (preserves the inner product), and defining ψ_{t} := e^{iAt}(ψ_{0}), we get

\frac{d}{dt}ψ_{t} = iAψ_{t}

This is why we need the Schrödinger operator to be a self-adjoint operator in a Hilbert space - primarily for obtaining solutions to Schrödinger’s equation! Which is what motivated Von Neumann and others to come up with the theory of Hilbert spaces and self-adjoint operators on them in the 20th century. Eventually, Kato and Rellich showed that the Ĥ in our setup above, is indeed self-adjoint.

But we would like a much more systematic theory of “quantizing” functions H : T^{∗}Q → ℝ and getting operators Ĥ : L^{2}(Q) → L^{2}(Q).

Even better, can we handle the case when the phase space X isn’t T^{∗}Q? (So we do not have Q - what would we replace L^{2}(Q) by?)

This leads us to “geometric quantization”. For more on this, try

  http://www.math.ucr.edu/home/baez/quantization.html

Then try Sniatycki’s book.

A lot of cohomology comes into the game - starting with the fact that [ω] ∈ H^{2}(X, ℝ) must come from an integral cohomology class, i.e. [ω] ∈ im φ, where φ : H^{2}(X, ℤ) → H^{2}(X, ℝ).

2. Jan 23, 2007: Categorification

Besides the “obvious” questions raised by the chart presented last time, there’s a bigger question: What’s really going on? Is the “quantization” some arbitrary trick (that nature has settled on), or does it have some deeper meaning? Let’s try to dig deeper!

2.1. A secret functor. What sort of entity is the action? Recall: in one approach, we have a configuration space Q, and then the action is a function S : P_{(t_{0},q_{0})→(t_{1},q_{1})} → ℝ, where if we denote x_{i} := (t_{i}, q_{i}) ∈ ℝ × Q, then the domain is more precisely written as

P_{x_{0}→x_{1}} := {γ : [t_{0}, t_{1}] → Q : γ(t_{i}) = q_{i}}

[figure: Q at both ends + path γ ∈ P_{x_{0}→x_{1}}]

But it’s much deeper than this! Note that the action respects composition of paths: given γ_{i} ∈ P_{x_{i−1}→x_{i}}, we can concatenate them to get a path γ_{1}γ_{2} ∈ P_{x_{0}→x_{2}}, and then

S(γ_{1}γ_{2}) = S(γ_{1}) + S(γ_{2})

What this secretly means is that the action is a functor, from some category 𝒫 with

• x = (t, q) ∈ ℝ × Q as objects

• given objects x_{0}, x_{1}, paths γ ∈ P_{x_{0}→x_{1}} as morphisms (so each P_{x_{0}→x_{1}} is a Hom-space in 𝒫)

• composition of morphisms given by concatenation of paths (this is indeed associative since we already have t in the “x-data”, and don’t need to reparametrize from [0, 1] → Q);

to some category ℝ with • one object, ∗

• all real numbers as morphisms

• composition of morphisms is just addition (a group is just a one object category, with all morphisms invertible).

Technical remarks:

(1) We cannot use all possible paths in defining 𝒫 (since the action and Lagrangian involved the derivative γ̇), nor only smooth paths (since composing paths might result in “corners”):

[figure: a concatenation of paths γ_{1}, γ_{2} with a “cusp”-like point of non-differentiability at the common point x_{1} = (t_{1}, q_{1})]

We could use piecewise smooth paths, though.

(2) Alternatively, we could have specified both the position and velocity (to avoid “corners”), but note that the calculus of variations involved in picking out an “extremal” path automatically chooses these by itself! So if we specified both postiion and velocity, then an extremal path might not exist. (Physically, we don’t want our particle/path to have “too much inherent data”.)

(3) Note that the above just talks of the action; we’ll now see how to use this categorical approach for classical and quantum mechanics. Thus, (in quantum mechanics) a path does not represent a particle’s (time) evolution, only (the integral over) the entire path space does! So, even though we do not know the positiion and momentum at the same time for a particle, we do know it for each specific path!

2.2. Bringing in arbitrary categories. So, can we do classical and quantum mechanics starting with any functor S : C → ℝ (where now C is any category), with

• “configurations” as objects, and

• “paths” as morphisms? To see this, we ask: how did we use S : C → ℝ in the chart last time?

Classically, we can do one of two things:

(1) We “criticize” it - i.e. for each x, y ∈ C, we look at S : Hom(x, y) = Hom_{C}(x, y) → ℝ and seek critical points, i.e. γ ∈ Hom(x, y) with dS(γ) = 0.

[figure: Q (or C) on both sides, with γ : x → y and δS(γ) = 0]

This only makes sense if each set Hom(x, y) is a manifold or a more general “infinite-dimensional manifold” (e.g. a space of piecewise smooth paths in a manifold Q), and S is differentiable.

This is addressed by the theory of smooth categories and smooth functors,

cf. http://www.math.ucr.edu/home/baez/2conn.pdf.

(1′) We “minimize” it - i.e. for each x, y ∈ C, we seek γ ∈ Hom(x, y) that minimizes S(γ). (We might need each Hom-space to be a topological space (possibly compact!), and S “continuous”.)

Remarks:

(1) For both (1) and (1′), the issues of existence and uniqueness of a γ criticizing/minimizing the action, are very important.

(2) Case (1′) is closer to quantum mechanics, which we can see if we study Hamilton’s principal function: given x, y ∈ C, define

Z(x, y) := inf_{γ∈Hom(x,y)} S(γ)

(assuming this infimum exists). In classical mechanics, this is very important; we get the Hamilton-Jacobi equations by differentiating Z(x, y) with respect to x or y, and working out the answer. These are the classical analogue(s) of Schrödinger’s equation.

Now consider the quantum case.

(2) We integrate its exponential - i.e. for each x, y, we compute a (transition) amplitude

Z_{ℏ}(x, y) := ∫_{γ∈Hom(x,y)} e^{iS(γ)/ℏ} 𝒟γ

For this, we want each set Hom(x, y) to be a measure space (or generalized measure space), and e^{iS(γ)/ℏ} must be integrable.

In this case, we can get Schrödinger’s equation by fixing x and differentiating Z_{ℏ}(x, y) with respect to y (or vice versa), and working out the answer.

Now compare cases (1′) (i.e. Z(x, y)) and (2) (i.e. Z_{ℏ}(x, y)): the classical and quantum cases.

Classical  Quantum

(a) S(γ) ∈ ℝ.

(a) e^{iS(γ)/ℏ} ∈ U(1) ⊂ ℂ (because integrating this gives us a number in ℂ, not U(1)).

(b) We take the infimum (i.e. minimum).

(b) We take the integral (i.e. sum).

(c) The group morphism is addition: S(γ_{1}γ_{2}) = S(γ_{1}) + S(γ_{2}).

(c) The group morphism is multiplication: e^{iS(γ_{1}γ_{2})/ℏ} = e^{iS(γ_{1})/ℏ} · e^{iS(γ_{2})/ℏ}.

Thus, we now ask: How are both of these, special cases of some “prescription” for getting physics out of the action?

3. Jan 30, 2007: Physics is rigged!

3.1. The analogous viewpoints. From last time, we’ve seen a “big analogy” between classical and quantum mechanics of point particles. As we saw, when we minimize/integrate, the identity for this operation in ℝ/ℂ (respectively) is +∞/0. Thus, we need to replace the ℝ used above, by ℝ^{min} := ℝ ∪ {+∞} throughout.

Moreover, since we add the actions of paths whenever we compose (in the classical case), we need to expand the addition operation to all of ℝ^{min}. This is done by: +∞ + x = +∞ for all x, and (ℝ^{min}, +, 0) is a commutative monoid.

We can now say more about the above analogy:

Classical mechanics of point particles Quantum mechanics of point particles

(a) We start with the

action S(γ) ∈ ℝ^{min}.

amplitude e^{iS(γ)/ℏ} ∈ ℂ.

(b) What we do:  take the infimum/minimum. take  the  inte-

gral/sum.

(c) Operational identity: +∞ - so (ℝ^{min}, min, +∞) is a commutative monoid.

0 - so (ℂ, +, 0) is an abelian group.

(d) When we compose paths, we add actions.

multiply amplitudes.

(e) Operational identity:

0.

1.

(f) Overall:

(ℝ^{min}, min, +∞, +, 0) is a commutative rig.

(ℂ, +, 0, ·, 1) is a commutative ring.

Remark 3.1.

(1) We’ll see later, how the picture on the right is really a one-parameter family (one for every ℏ), and we “go to the classical case” as ℏ → 0.

(2) A \underline{rig} is a “\underline{ring} without \underline{n}egatives”, i.e. a commutative monoid under addition, and a monoid under multiplication, satisfying left/right distributive laws. (In particular, every ring is a rig.)

(3) ℝ^{min} is a rig that satisfies x min x = x for all x. Thus, it satisfies an “idempotence” property, sort of the opposite of the usual “cancellational” addition that we see in a group (a ∗ b = a ∗ c ⇒ b = c.)

3.2. Switching between the classical and quantum viewpoints. In the rest of this lecture, we’ll go back and forth between the classical and quantum viewpoints. For instance,

(1) In classical mechanics, action is a functor S : C → ℝ^{min}, where C is any category (whose objects - resp. morphisms - are called “configurations” - resp. “paths”), and ℝ^{min} is a category with one object whose morphisms are x ∈ ℝ^{min} and composition is addition (i.e. the “multiplication” in the rig): S(γ_{1}γ_{2}) = S(γ_{1}) + S(γ_{2}).

Let’s now “quantize” this statement!

In quantum mechanics, amplitude is a functor e^{iS(·)/ℏ} : C → ℂ, where C is as above, and ℂ is a category with one object whose morphisms are x ∈ ℂ and composition is ·: e^{iS(γ_{1}γ_{2})/ℏ} = e^{iS(γ_{1})/ℏ} · e^{iS(γ_{2})/ℏ}.

Remark 3.2. The amplitude functor is called e^{iS(·)/ℏ} more for sentimental reasons here, than out of any connection with S itself.

(2) We now present an example, wherein we “translate” a concept (over to the “other side”) using this analogy. However, unlike earlier, we now start on the “quantum side”!

Say C = 𝒫 (as in the last class), with

• objects as points x = (t, q) ∈ ℝ × Q, where Q is some “configuration space” (manifold),

• morphisms γ : x_{0} = (t_{0}, q_{0}) → x_{1} = (t_{1}, q_{1}) are paths : [t_{0}, t_{1}] → Q so that γ(t_{i}) = q_{i}.

In the quantum case, a wavefunction ψ : Q → ℂ tells us the amplitude for a particle to be at q ∈ Q. Of course, this is not a “good” thing because we have Q (and not ℝ × Q), and this is not really among the objects or morphisms of our category C.

But now, we can bring in C as follows: We describe the timeevolution of ψ by

ψ(t_{1}, q_{1}) = ∫_{q_{0}∈Q} ∫_{P_{q_{0}→q_{1}}} e^{iS(γ)/ℏ}ψ_{t_{0}}(q_{0})(𝒟γ) dq_{0}

The classical analogue of a wavefunction is a known entity in physics; let’s call it ψ_{c} for short. (Just as S(·) ↦ e^{iS(·)/ℏ}, we really should have ψ in the quantum case coming from −iℏ ln ψ, but we just use ψ_{c}.)

Thus, classically, ψ_{c} : Q → ℝ^{min} tells us the action for a particle to be at q ∈ Q. By our analogy, it should evolve in time as follows:

ψ_{c}(t_{1}, q_{1}) = inf_{q_{0}∈Q} inf_{γ:[t_{0},q_{0}]→[t_{1},q_{1}]}(S(γ) + ψ_{c}(t_{0}, q_{0}))

Remark 3.3.

(a) Note that t_{0} is fixed.

(b) The 𝒟γ · dq_{0} now just gives information / identifies the space over which we integrate / minimize.

(c) If we imagine ψ_{c}(t_{0}, q_{0}) as the cost to “start a trip” at q_{0} ∈ Q_{0} at time t_{0}, and S(γ) as the cost of the trip γ, this formula tells us that ψ_{c}(t_{1}, q_{1}) is the cheapest price to be at q_{1} ∈ Q at time t_{1}. (Note that there may be many different “equally cheap” ways to get there!)

(3) In the quantum case, you can go ahead and use the path integral to

compute \frac{d}{dt}ψ - you get Schrödinger’s equation. In the classical case, you get the Hamilton-Jacobi equations.

3.3. Wick rotation and a spring in imaginary time (revisited). Last quarter, we saw that the dynamics of point particles is analogous to the statics of strings. This analogy involves Wick rotation, namely, the substitution t ↦ −it. (Think of how “rotating” clockwise by 90^{∘} amounts to multiplying by −i in the complex plane.)

For example, consider a rock and a spring in a gravitational field:

[figure: rock and spring in gravitational fields - give parabolic shapes oriented oppositely]

The action for the rock is

S = ∫_{t_{0}}^{t_{1}} [\frac{m}{2} q̇ · q̇ − V (q(t))] dt

which, under Wick rotation, becomes (note that q̇ = \frac{d}{dt}q, so we have to bring in −1 from the q̇ · q̇ now)

∫_{−it_{0}}^{−it_{1}} [−\frac{m}{2} q̇ · q̇ − V (q(−it))] d(−it)

for the spring. This was what we referred to as the “spring in imaginary time” early last quarter! In this (second/spring) case, we write it as the energy (cancelling various powers of (−i) and renaming variables):

E = −iS = ∫_{t_{0}}^{t_{1}} [\frac{m}{2} q̇ · q̇ + V (q(t))] dt

since this is the energy of the spring, where m is now the tension (spring constant), q̇ refers to how “stretched” the spring is, and the energy is the sum of the “tension energy” and the gravitational (potential) energy.

Next time: The rig ℝ (not ℝ^{min}) comes into play!

4. Jan 23, 2007: Statistical mechanics and deformation of rigs

We saw last time that the classical mechanics (dynamics) of particles becomes the classical statics of strings by doing the substitutions

t ↦ −it, S ↦ iE

Minimizing action now becomes minimizing energy.

What does the quantum mechanics (dynamics) of particles become when we do these substitutions?

4.1. Statistical mechanics “quantizes” strings. In quantum mechanics, the relative amplitude for a particle to trace out a path is e^{iS()/ℏ}. In “statistical” mechanics (really thermal statics - classical statics but with nonzero temperature T) - and let us not even talk of thermodynamics now!

- the relative probability for a system to be in configuration of energy E is

e^{−E/kT}

where k is Boltzmann’s constant (a conversion factor between energy and temperature).

Note that we have

e^{iS/ℏ} ↦ e^{−E/kT}

if we do the substitutions

S ↦ iE, ℏ ↦ kT

(or should we really have S ↦ iE/T, because ℏ is a constant? But we also want to get to classical mechanics by ℏ → 0, or T ↦ 0!).

This makes some sense since ℏ measures how big “quantum fluctuations” are:

[figure: quantum (“wiggling”) version of a thrown rock]

(Thus, all paths “far” from the path of least action cancel one another out, and only the “nearby” paths contribute.)

while kT measures how big “thermal fluctuations” are:

[figure: hanging spring at nonzero (positive!) temperature] (Moreover, the relative quantities need to be normalized to give the amplitude/probability.)

4.2. A family of rigs via the Boltzmann map. Henceforth I’ll set k = 1 and use the substitution ℏ ↦ T. Note that

• e^{iS/ℏ} ∈ ℂ, the rig of relative amplitudes, and

• e^{−E/T} ∈ ℝ^{+}, the rig of relative probabilities.

(Here, ℝ^{+} = ([0, ∞), +, 0, ·, 1).) In short, we have

Particles (p = 1)

Strings (p ≥ 0)

(1) Classical dynamics - we deal with action S ∈ ℝ^{min} (1) Classical statics - we deal with energy E ∈ ℝ^{min} (a “different/imaginary” ℝ^{min})

(2) Quantum dynamics - we deal with relative amplitude e^{iS/ℏ} ∈ ℂ (2) Thermal statics - we deal with relative probability e^{−E/T} ∈ ℝ^{+}

We note that to go from the first column to the second, we use Wick rotation:

t ↦ −it, S ↦ iE, ℏ ↦ T

We’d like to understand how quantum mechanics reduces to classical mechanics as ℏ → 0, but it’s easier to understand how thermal statics reduces to classical statics as T → 0.

To do this, we’ll formulate thermal statics using E instead of e^{−E/T}: for any T > 0, we consider the Boltzmann map β_{T} : ℝ^{min} → ℝ^{+}:

E ↦ e^{−E/T} (+∞ ↦ 0)

This isn’t a rig homomorphism, just a one-to-one and onto function. So, we’ll pull back the rig structure on ℝ^{+} to (the set) ℝ^{min} via β_{T}, and get a rig ℝ^{T}.

As a set, ℝ^{T} is just [0, ∞), but now it’s a rig with

a +_{T} b := β_{T}^{−1}(β_{T} (a) + β_{T} (b))
0_{T} := β_{T}^{−1}(0)
a ·_{T} b := β_{T}^{−1}(β_{T} (a)β_{T} (b))
1_{T} := β_{T}^{−1}(1)

Homework. Work out +_{T} , 0_{T} , ·_{T} , 1_{T} explicitly, and show that

lim_{T →0} β_{T}^{−1}(ℝ^{+}) = ℝ^{min}

or, in other words, that

lim_{T →0}(+_{T} , 0_{T} , ·_{T} , 1_{T}) = (min, +∞, +, 0)

So, the “topological rig” ℝ^{T} converges to the topological rig ℝ^{min} as T → 0.

Also note that as T → +∞, all elements β_{T} (a) converge to either 0 (if a ∈ ℝ) or 1 (if a = +∞) - “impossible” or “possible” events respectively. (That is, when the going gets hot, everything that is possible appears equally likely!)

(We’d be very happy if this were to be in a different rig - the logic rig of truth values ({0, 1}, ∨, ∧). But this is not so.)

4.3. The analogous situation for quantization. The moral of the above analysis is that “thermal statics reduces to classical statics as T → 0”; in both cases we’re really doing linear algebra over some rig, and ℝ^{T} → ℝ^{min} as T → 0.

Alas, seeing classical mechanics as an ℏ → 0 limit of quantum mechanics is harder, since

β_{ℏ} : ℝ^{min} → ℂ sending S ↦ e^{iS/ℏ}, +∞ ↦ 0

is neither one-to-one nor onto, and its image is not a subrig (though it’s closed under multiplication). So we can’t pull the rig structure on ℂ back to ℝ^{min}.

However, people do study quantization indirectly using the lim_{T →0} ℝ^{T} =

ℝ^{min} idea, which is called

• tropical mathematics (a really stupid term for the work of Brazilian mathematicians - “Arctic mathematics” would be better for T → 0 math!)

• idempotent analysis (since a min a = a in ℝ^{min})

• Maslov dequantization (in reference to how T → 0 limit lets us study the ℏ → 0 limit).

Solution to the homework. It is easy to see the following:

β_{T}^{−1}(a) := −T ln(a)
a +_{T} b := −T ln(e^{−a/T} + e^{−b/T})
0_{T} := −T ln 0 = +∞
a ·_{T} b := −T ln(e^{−(a+b)/T}) = a + b
1_{T} := −T ln 1 = 0

This means that there is only one limit left to verify: that of a +_{T} b as T → 0. So say a ≤ b. Then we compute:

a +_{T} b = −T ln(e^{−a/T} (1 + e^{(a−b)/T})) = −T ln(e^{−a/T}) − T ln(1 + e^{(a−b)/T})

The first term clearly equals a. Now denote α = e^{a−b}. Then α ∈ (0, 1]. As (1 >)T → 0^{+}, we see that α^{1/T} ∈ (0, 1], whence ln(1 + α^{1/T}) is bounded. Therefore, by the Pinching Theorem,

lim_{T →0} a +_{T} b = a + lim_{T →0^{+}} T · ln(1 + α^{1/T}) = a + 0

and we are done, since a min b = a. □

5. Feb 13, 2007: An example of path integral quantization - I

We have a strategy for quantization, given any category C (of “configurations” and “paths”) and a functor

S : C → (ℝ, +)

(the “action”). This gives a functor

e^{iS/ℏ} : C → (ℂ, ·)

(the “amplitude”), and we compute the “transition amplitude” from any object x ∈ C to y ∈ C via

Z_{ℏ}(x, y) = ∫_{γ:x→y} e^{iS(γ)/ℏ}𝒟γ

which requires also that we have measures on hom-sets hom_{C}(x, y).

5.1. Example: free particle on the real line. Let’s do an example - the free particle on ℝ. Here, the objects of C form the set ℝ^{2} ∋ (t, q), and morphisms γ : (t_{0}, q_{0}) → (t_{1}, q_{1}) are paths γ : [t_{0}, t_{1}] → ℝ, so that γ(t_{i}) = q_{i}.

[figure: path γ from x_{0} = (t_{0}, q_{0}) to x_{1} = (t_{1}, q_{1})]

(This is the category 𝒫 - with Q = ℝ - that we introduced earlier, and then the morphism spaces {γ : x_{0} → x_{1}} were called P_{q_{0}→q_{1}}.) Thus,

Z_{ℏ}((t_{0}, q_{0}), (t_{1}, q_{1})) = ∫_{P_{q_{0}→q_{1}}} e^{iS(γ)/ℏ} 𝒟γ

where S is the action for a free (i.e. no potential) particle of mass m:

S(γ) = ∫_{t_{0}}^{t_{1}} L(γ(t), γ̇(t)) dt

with Lagrangian given by

L(q, q̇) = \frac{m}{2}q̇^{2}

since there’s no potential.

To do the path integral over all paths γ, we first integrate only over piecewise linear paths (and worry about “taking the limit” later). However, we first need a change of notation!

(t_{0}, q_{0}) ↔ (t, q), (t_{1}, q_{1}) ↔ (t′, q′)

We thus consider piecewise linear paths γ : (t, q) → (t′, q′)

[figure: zigzagging path with corners at (t_{j}, x_{j}) for j = 2, 3, …, N]

for some chosen times t_{1}, …, t_{N+1}, where

P_{t} := {t = t_{1} < t_{2} < ⋯ < t_{N} < t_{N+1} = t′}

is a (not necessarily regular) partition of [t, t′] (also called a mesh), and γ is piecewise linear on each subinterval.

To integrate over all these piecewise linear paths, we just integrate over x_{2}, …, x_{N} ∈ ℝ, where x_{j} = γ(t_{j}).

Then we’ll try to show that these integrals over (the space of) piecewise linear paths converge as the norm ||P_{t}|| of the partition goes to zero. To use another name for the norm, we’ll see what happens as the mesh spacing

max_{j}(t_{j+1} − t_{j})

goes to zero.

5.2. Doing the math. But first, let’s see what these integrals look like - let’s compute one:

A_{P_{t}} = A_{P_{t}}((t, q), (t′, q′)) = ∫_{ℝ^{N−1}} exp (\frac{i}{ℏ} ∫_{t}^{t′} \frac{m}{2}γ̇(s)^{2} ds) dx_{2} … dx_{N}

Actually, we need to rescale the Lebesgue measure by normalizing factors:

dx_{j} ↦ \frac{dx_{j}}{c_{j}} , c_{j} ∈ ℝ

where c_{j} depends on t_{j+1} − t_{j} (for all j). We need these to get convergence as the mesh spacing goes to zero.

Question. The c_{j}’s are chosen to make the math work out fine. (These are what one calls (re?)normalizations.) But what is the physics behind choosing them? (E.g. is it just analogous to rescaling in order to get the total probability to equal 1?) Many people would be very happy to know...

But γ is piecewise-linear, so on the jth piece [t_{j}, t_{j+1}], we have

γ̇(s) ≡ \frac{x_{j+1} − x_{j}}{t_{j+1} − t_{j}} = \frac{Δx_{j}}{Δt_{j}}

where Δx_{j} = x_{j+1} − x_{j}, Δt_{j} = t_{j+1} − t_{j} for all j. Hence,

A_{P_{t}} = ∫_{ℝ^{N−1}} exp (\frac{im}{2ℏ} ∑_{j=1}^{N} (\frac{Δx_{j}}{Δt_{j}} )^{2} Δt_{j}) \frac{dx_{2}}{c_{2}} … \frac{dx_{N}}{c_{N}}

The crucial thing is that if we choose the c_{j}’s correctly, A_{P_{t}} is actually independent of the mesh/partition P_{t} - so convergence is trivial!

In other words, we can actually compute Z_{ℏ}((t, q), (t′, q′)) as an integral over linear paths

[figure: a straight line path between (t, q) and (t′, q′)]

of which there is only one.

To prove that A_{P_{t}} is independent of the mesh P_{t}, let’s think instead about the rule for evolving a wavefunction ψ in time:

[figure: wavefunctions ψ_{t}, ψ_{t′} : Q → ℂ, and a possible path from (t, q) to (t′, q′)]

ψ_{t′}(q′) = ∫_{ℝ} ∫_{γ:(t,q)→(t′,q′)} e^{iS(γ)/ℏ}ψ_{t}(q) 𝒟γ dq = ∫_{ℝ} Z_{ℏ}((t, q), (t′, q′))ψ_{t}(q) dq

When we approximate Z_{ℏ} by integrating over piecewise linear paths, we get (ψ̃_{t} is the approximation, and we write out the exponentials in the order in which they are applied to the original wavefunction ψ_{t}(q)):

ψ̃_{t′}(q′)
= ∫_{ℝ^{N}} exp (\frac{im}{2ℏ} \frac{(Δx_{N})^{2}}{Δt_{N}} ) ⋯ exp (\frac{im}{2ℏ} \frac{(Δx_{1})^{2}}{Δt_{1}} ) ψ_{t}(q) dq \frac{dx_{2}}{c_{N}} ⋯ \frac{dx_{N}}{c_{N}}
= ∫_{ℝ^{N}} K(Δt_{N}, Δx_{N}) ⋯ K(Δt_{1}, Δx_{1})ψ_{t}(q) dq dx_{2} ⋯ dx_{N}

where

K(Δt_{j}, Δx_{j}) := \frac{1}{c_{j}} exp (\frac{im}{2ℏ} \frac{(Δx_{j})^{2}}{Δt_{j}} )

is the “kernel”.

Remark 5.1.

(1) We have to justify the third equality in

ψ_{t′}(q′) = ∫_{ℝ^{N}} Z_{ℏ} · ψ_{t}(q) = ∫_{ℝ^{N}} lim_{||P_{t}||→0} A_{P_{t}} · ψ_{t}(q)
= lim_{||P_{t}||→0} ∫_{ℝ^{N}} A_{P_{t}} · ψ_{t}(q) = lim_{||P_{t}||→0} ψ̃_{t′}(q′)

But if A_{P_{t}} is independent of P_{t}, then the equality becomes obvious! Moreover, we shall see next time that this does, indeed, hold.

(2) To show that A_{P_{t}} is independent of P_{t}, we just need to show that ψ̃_{t′}(q′) is independent of P_{t} for all ψ_{t}. (This is exactly like showing that ∫ fg = 0 ∀g ⇒ f = 0, for then we can choose various deltafunctions for ψ_{t}, and get that ψ̃_{t′}(q) = 0 for all q.)

(3) To show independence from P_{t}, it is enough to show that if we refine the mesh/partition by one point, we get the same kernel. This is because if we are then given A_{P} , A_{Q} for partitions P, Q, then they are both equal to A_{P ∪Q}.

Thus, it is now enough to show that

K(t_{3} − t_{1}, x_{3} − x_{1}) = ∫_{x_{2}∈ℝ} K(t_{3} − t_{2}, x_{3} − x_{2})K(t_{2} − t_{1}, x_{2} − x_{1}) dx_{2}

[figure: γ is a line, but we introduce an intermediate time t_{2} and a non-linear, piecewise-linear curve from t_{1} to t_{3}]

(4) Note that this is a proof where the c_{j}’s are unknown! So we will in fact use the proof to determine the c_{j}’s (so that the above condition holds). Moreover, we need c_{j} to be a function of at most the time - and as we will see next time, c_{j} actually depends only on Δt_{j}!

6. Feb 20, 2007: An example of path integral quantization - II

Last time, we considered path integral quantization for a free particle on a line. (This is “exactly solvable”.) We claimed that the exact answer could be found by considering just one path - the straight line path

[figure: straight line from (t, x) to (t′, x′)]

- and essentially evaluating e^{iS/ℏ} to go from (t, x) to (t′, x′):

K(Δt, Δx) = \frac{1}{c(Δt)} exp (\frac{im}{2ℏ} \frac{(Δx)^{2}}{Δt} )

where c = c(Δt) is a normalizing factor (that we have yet to determine). We saw that to prove this, we just need to check

K(t_{3} − t_{1}, x_{3} − x_{1}) = ∫_{x_{2}∈ℝ} K(t_{3} − t_{2}, x_{3} − x_{2})K(t_{2} − t_{1}, x_{2} − x_{1}) dx_{2} (⋆)

i.e., in pictures,

[figure: Feynman diagram line = ∫ ( Feynman diagram piecewise line bro-

ken at (t_{2}, x_{2})) dx_{2}]

6.1. Time-evolution operators. To prove (⋆) (for some c(Δt)), we could just do the integral - but this is too annoying (for JB!). So instead, we’ll take a more conceptual route:

Consider the operator U(t) which describes one step of time evolution (via straight-line paths only):

[figure: wavefunctions at times t_{1}, t_{2}; the endpoint is fixed at (t_{2}, x_{2}), but there are lots of lines leading to it, all from potential starting points (t_{1}, x_{1})]

(U(t_{2} − t_{1})ψ_{t_{1}}) (x_{2}) = ∫_{ℝ} K(t_{2} − t_{1}, x_{2} − x_{1})ψ_{t_{1}} dx_{1}

This tells us the wavefunction at time t_{2} in terms of the wavefunction at time t_{1} (i.e. ψ_{t_{1}}), as an integral over straight-line paths from (t_{1}, x_{1}) to (t_{2}, x_{2}).

Remark 6.1. Thus, kernels really are functions of two variables that are like matrices, but with entries indexed by a continuous set. Integrating against a kernel is like matrix multiplication - see the last two equations above!

In these (above) terms, (⋆) simply says

U(t_{3} − t_{1})ψ_{t_{1}} ≡ U(t_{3} − t_{1})U(t_{2} − t_{1})ψ_{t_{1}} ∀ψ_{t_{1}}, ∀t_{i} ∈ ℝ

or equivalently,

U(t + s) ≡ U(t)U(s) ∀t, s ∈ ℝ

(One way is clear: integrate the second identity against any ψ_{t_{1}} to get the first. Conversely, the familiar principle of ∫ fg = 0 ∀g ⇒ f ≡ 0 suggests that we use delta-functions in place of ψ_{t_{1}} to get the second equation at all points.)

6.2. Bringing in the Hamiltonian. We would know this if we could write

U(t) ≡ exp (\frac{−itH}{ℏ} ) = exp (\frac{t}{iℏ}H)

for some operator H, since we then get

exp (\frac{−i(t + s)H}{ℏ} ) ≡ exp (\frac{−itH}{ℏ} ) · exp (\frac{−isH}{ℏ} )

So we will show that U(t) = exp((t/iℏ)H) for

H = −\frac{ℏ^{2}}{2m} \frac{d^{2}}{dx^{2}}

(as Schrödinger had obtained). Here, H is the Hamiltonian for the free particle.

When we show this, we’ll see that if ψ_{t} = U(t)ψ_{0}, then ψ_{t} will satisfy Schrödinger’s equation:

\frac{d}{dt}ψ_{t} = \frac{d}{dt} [exp (\frac{t}{iℏ}H) ψ_{0}] = \frac{1}{iℏ}H (e^{(t/iℏ)H}ψ_{0}) = \frac{1}{iℏ}Hψ_{t}

So we just need to check that (e^{(t/iℏ)H}ψ)(x) is the same as

(U(t)ψ)(x) = ∫_{y∈ℝ} K(t, x − y)ψ(y) dy

Since both sides depend linearly on ψ, and both sides are translationinvariant, it suffices to check this for ψ = δ, the Dirac-delta at the origin. In other words,

(e^{(t/iℏ)H}δ) (x) = K(t, x) = K(t − 0, x − 0)

Remark 6.2.

(1) To make this rigorous, we need to • introduce some topology on the space of wavefunctions,

• show that in this space, the set of finite linear combinations of delta functions is “dense” in this topology, and

• show that exp(\frac{t}{iℏ}H), U(t) are “continuous” on this space in this topology.

(2) Note that in this last equation,

• the left side is the Hamiltonian way of computing the amplitude for a particle to end up at position x at time t if it (definitely!) starts at the origin at time 0; and

• the right side is the Lagrangian way to compute the same thing - but integrating only over straight-line paths!

6.3. Computing the normalizing factors. Since we (almost) know one side - K(t, x) - we’ll just compute the other side of the above equation. We’ll use the Fourier Transform

ψ̂(k) = \frac{1}{\sqrt{2π}} ∫_{ℝ} e^{−ikx}ψ(x) dx (k ∈ ℝ)

and its inverse

ψ(x) = \frac{1}{\sqrt{2π}} ∫_{ℝ} e^{ikx}ψ̂(k) dk

(Here, “k” stands for momentum.)

Next, we need a couple of small computations. Firstly, how do differentiation and the Fourier transform interact? We use integration by parts to compute:

ψ̂′(k) = \frac{1}{\sqrt{2π}} ∫_{ℝ} e^{−ikx} \frac{d}{dx}ψ(x) dx = ik \frac{1}{\sqrt{2π}} ∫_{ℝ} e^{−ikx}ψ(x) dx = ikψ̂(k)

so that for H = −\frac{ℏ^{2}}{2m} \frac{d^{2}}{dx^{2}} , we get (using the Taylor series expansion)

{ \atop e^{(t/iℏ)H}ψ}(k) = { \atop e^{\frac{−t}{iℏ} \frac{ℏ^{2}}{2m} \frac{d^{2}}{dx^{2}}} ψ}(k) = ∑_{n≥0} \frac{1}{n!} (\frac{itℏ}{2m})^{n} { \atop \frac{d^{2n}}{dx^{2n}} ψ}(k)

= ∑_{n≥0} \frac{1}{n!} (\frac{itℏ · i^{2}k^{2}}{2m} )^{n} ψ̂(k) = e^{−itk^{2}ℏ/2m}ψ̂(k)

To simplify future computations, let’s pick units where ℏ = 1 and m = 1, to lessen the mess. So we now know that e^{(t/iℏ)H} now becomes e^{−itH}, and we compute:

{ \atop e^{−itH}δ}(k) = e^{−itk^{2}/2}δ̂(k) = e^{−itk^{2}/2} \frac{1}{\sqrt{2π}} ∫_{ℝ} e^{−ikx}δ(x) dx = \frac{e^{−itk^{2}/2}}{\sqrt{2π}}

where we compute δ̂(k) from first principles.

Now take the inverse Fourier transform:

(e^{−itH}δ)(x) = \frac{1}{2π} ∫_{ℝ} e^{ikx}e^{−itk^{2}/2} dk
= \frac{1}{2π} ∫ exp [− (\frac{i}{2} (tk^{2} − 2xk + \frac{x^{2}}{t} ) − \frac{i}{2} \frac{x^{2}}{t} )] dk

= \frac{e^{ix^{2}/2t}}{2π} ∫ exp [\frac{−i}{2} (\sqrt{t}k − \frac{x}{\sqrt{t}})^{2}] dk

= \frac{e^{ix^{2}/2t}}{2π\sqrt{t}} ∫ exp (\frac{−i}{2}u^{2}) du

where we use the u-substitution u = \sqrt{t}k − \frac{x}{\sqrt{t}} in the last step - and the last integral is just a number.

So just as desired, (e^{−itH}δ)(x) = K(t, x), where K(t, x) = \frac{e^{\frac{i}{2} \frac{x^{2}}{t}}}{c(t)} , and the

normalizing factor is

\frac{1}{c(t)} = \frac{1}{2π} \frac{1}{\sqrt{t}} ∫_{ℝ} exp (\frac{−i}{2}u^{2}) du

Note that this integral is not always absolutely convergent (this is similar, for instance, to the fact that a series does not converge if its summand terms do not go to zero). However, to evaluate it, we can consider lim_{R→∞} ∫_{−R}^{R}, or alternatively, consider

∫ e^{−τu^{2}/2} du

where τ ∈ ℂ is close to i, but has a small positive real part. This then converges absolutely; we then take the limit as τ → i. Let’s do it:

∫_{ℝ} e^{−τu^{2}/2} du = \sqrt{∫_{ℝ} e^{−τx^{2}/2} dx · ∫_{ℝ} e^{−τy^{2}/2} dy}

= \sqrt{∫∫_{ℝ^{2}} e^{−τ(x^{2}+y^{2})/2} dx dy}

Now convert to polar coordinates, and the Jacobian is r (this procedure is very well-known), and compute

= \sqrt{∫_{0}^{∞} ∫_{0}^{2π} e^{−τr^{2}/2}r dθ dr} = (v = \frac{r^{2}}{2} ) = \sqrt{2π ∫_{0}^{∞} e^{−τv} dv} = \sqrt{\frac{2π}{τ}}

We now take the limit as τ → i. Thus,

K(t, x) = \frac{e^{ix^{2}/2t}}{2π\sqrt{t}} · \sqrt{\frac{2π}{i}} = \frac{e^{ix^{2}/2t}}{\sqrt{2πit}}

This is how to do the “simplest path-integral in the world”!

Next time: We throw in a potential - then it may not be exactly solvable, but we can still go some of the distance.

7. Feb 27, 2007: More examples of path integrals

7.1. A potential problem. Now let’s consider a quantum particle of mass m on the real line, but in a potential

V : ℝ → ℝ

Let’s compute its time evolution using a path integral, using the fact that we already did (this for) the free particle. Our particle can trace out any path

γ : [0, T] → ℝ

(where now, without loss of generality, we start the clock ticking at 0), and its action is

S(γ) = ∫_{0}^{T} (\frac{m}{2}γ̇(t)^{2} − V (γ(t))) dt

so the path integral philosophy tells us:

Given a wavefunction ψ_{0} ∈ L^{2}(ℝ) at time 0, it will evolve to ψ_{T} ∈ L^{2}(ℝ) at time T, where

ψ_{T} (x′) = ∫_{x∈ℝ} ∫_{γ∈P_{x→x′}} e^{iS(γ)/ℏ}ψ_{0}(x) 𝒟γ dx

(Here, P_{x→x′} = {γ : [0, T] → ℝ, γ piecewise regular, γ(0) = x, γ(T) = x′}.)

To do this, we first integrate over piecewise linear paths like

[figure: piecewise linear path with corners at x = x_{0}, x_{1}, …, x_{n} = x′, but with timepoints t_{i} in a regular partition of [0, T]]

(Here, Δt = T/n, and we only consider regular partitions of [0, T]. We now set x_{k} = γ(kΔt), Δx_{k} = x_{k} − x_{k−1} whenever defined.)

We now take the limit n → ∞, if possible. So we hope

ψ_{T} (x′) = lim_{n→∞} ∫_{ℝ^{n}} e^{iS(γ)/ℏ}ψ_{0}(x_{0})\frac{dx_{0}}{c(Δt)} ⋯ \frac{dx_{n−1}}{c(Δt)}

where

S(γ) = ∫_{0}^{T} (\frac{m}{2}γ̇(t)^{2} − V (γ(t))) dt ∼ ∑_{k=1}^{n} (\frac{m}{2} (\frac{Δx_{k}}{Δt} )^{2} Δt − V (x_{k−1})) Δt

where the last expression merely an approximation, not an equality. Thus,

we have decided to approximate ∫_{(k−1)Δt}^{kΔt} V (γ(t)) by V (x_{k−1}). (Note that

as n → ∞, this approximation might not matter!)

So, we hope:

ψ_{T} (x′) = lim_{n→∞} ∫_{ℝ^{n}} ∏_{k=1}^{n} e^{\frac{im}{2ℏ} \frac{(Δx_{k})^{2}}{Δt}} e^{\frac{−i}{ℏ}V (x_{k−1})Δt} · ψ_{0}(x_{0})\frac{dx_{0}}{c(Δt)} ⋯ \frac{dx_{n−1}}{c(Δt)}

Hence we start with our wavefunction ψ_{0}, and repeatedly applying the following two types of operators in alternation:

(1) multiplication by exp (\frac{−i}{ℏ}V Δt), and

(2) evolving it for a time Δt as if it were a free particle.

The latter step, we’ve seen, amounts to the operator

ψ ↦ e^{−iHΔt/ℏ}ψ

where H is the Hamiltonian for a free particle:

H = −\frac{ℏ^{2}}{2m} \frac{d^{2}}{dx^{2}}

So ψ_{T} = lim_{n→∞} (e^{−iHΔt/ℏ}e^{−iV Δt/ℏ})^{n} ψ_{0}.

7.2. The Lie-Trotter Theorem and self-adjoint operators. We now need the following theorem.

Theorem 7.1 (Lie-Trotter). If A and B are (possibly unbounded) selfadjoint operators defined on a Hilbert space ℋ (with dense domains D(A), D(B) respectively), so that A+B is essentially self-adjoint on D(A)∩D(B), then

e^{i(A+B)t}ψ = lim_{n→∞}(e^{iAt/n}e^{iBt/n})^{n}ψ

for all ψ ∈ ℋ .

Remark 7.2.

(1) Unbounded self-adjoint operators are only “densely defined” on ℋ . Thus, A + B is only necessarily defined on D(A) ∩ D(B).

(2) Now, if A is such an operator, then for each t, e^{iAt} is unitary. We can thus define e^{iAt/n}e^{iBt/n} on D(A) ∩ D(B), and the definition can be extended (for this operator) to all of ℋ .

(3) For details, see Volume 1, Methods in Modern Mathematical Physics, by Reed and Simon.

In fact, H and V (i.e. mult_{V}) are self-adjoint on L^{2}(ℝ), and H + V is essentially self-adjoint on D(H) ∩ D(V), if V is reasonably nice - e.g. continuous and bounded below. So in this case,

ψ_{T} = lim_{n→∞} (e^{−iHΔt/ℏ}e^{−iV Δt/ℏ})^{n} ψ_{0}

exists and equals e^{−i(H+V)T/ℏ}ψ_{0}. So we get that ψ_{T} = e^{−i(H+V)T/ℏ}ψ_{0}, and if ψ_{0} ∈ D(H + V), we can differentiate this and get (set T ↔ t):

\frac{d}{dt}ψ_{t} = \frac{−i}{ℏ}(H + V)ψ_{t}

which is Schrödinger’s equation.

7.3. Generalization to complete Riemannian manifolds. We can also handle the case of a particle on a complete (connected) Riemannian manifold Q (see the lecture on January 16, 2007). Here again,

H = −\frac{ℏ^{2}}{2m}∇^{2}

and “V ” = mult_{V} (V : Q → ℝ) are self-adjoint operators on L^{2}(Q), and if V is continuous and bounded below, H + V is essentially self-adjoint on D(H) ∩ D(V).

Remark 7.3.

(1) The notion of piecewise linear paths does not make sense here. But we can still go for “piecewise geodesic” paths, each piece in a small enough coordinate-patch.

(2) Then, e^{−iHT/ℏ} is not exactly an integral over such paths with only a few pieces; we need to take the limit as the number n of pieces goes to infinity.

(3) So again, the final answer is as desired, but only in the limit - so the intermediate steps are only approximations now, unlike the case Q = ℝ.

(4) Some people study this after applying Wick rotation; this leads to the study of the Heat equation and of Brownian motion on manifolds.

Upshot. So again, skipping lots of steps, we obtain this formula for the time evolution of a wavefunction for a particle on Q with potential V:

ψ_{T} = lim_{n→∞} (e^{−iHT/nℏ}e^{−iV T/nℏ})^{n} ψ_{0} = e^{−i(H+V)T/ℏ}ψ_{0}

where H + V ↔ −\frac{ℏ^{2}}{2m}∇^{2}+ mult_{V}. (See the notes from January 16, 2007!)

Here, the operator ∇^{2} is defined for general Q, and H +V is the Hamiltonian for our particle.

7.4. Back to the general picture. Now let’s return to our general story. We have a category C whose objects are “configurations” and whose morphisms are “paths”. In the example we just saw, objects were points in ℝ×Q (spacetime) and a morphism γ : (t, x) → (t′, x′) is a path γ : [t, t′] → Q so that γ(t) = x, γ(t′) = x′.

[figure: γ : (t, x) → (t′, x′)]

We also have a functor S : C → ℝ (where ℝ is the category with one object, reals as morphisms, + as composition) serving as our action, giving

e^{iS/ℏ} : C → U(1) ⊂ ℂ

Question. How do we get a Hilbert space from this general framework? In the examples we just saw, we could use L^{2}(Q) - but there’s no “Q” in general! Q came from our ability to “slice” he set of objects ℝ × Q into slices {t = constant}, called Cauchy surfaces - surfaces on which we can freely specify “initial data” (= Cauchy data) for our wavefunction (and then we can solve differential equations given such Cauchy data - for example, Schrödinger’s equation).

7.5. Digression of the day: Cauchy surfaces. In Newtonian mechanics, spacetime is ℝ × ℝ^{3}, so Cauchy surfaces are the level sets for the first coordinate.

[figure: parallel family of horizontal straight lines]

In special relativity, spacetime is ℝ^{4} with the Minkowski metric. So Cauchy surfaces can be one of many different families of parallel lines. Lorentz transformations take one family of Cauchy surfaces to another.

[figure: parallel families of horizontal straight lines, cutting one another transversely]

Finally, in general relativity, spacetime is a 4-manifold with a Lorentzian metric; then Cauchy surfaces can be very badly behaved. They may not even exist! For instance, consider S^{1} × ℝ^{3}. There are closed timelike loops here.

(In fact, the problem in many (most?) time-traveller paradoxes in science fiction, is that of a lack of Cauchy surfaces. You cannot be in two places at once in the same universe!)

8. Mar 6, 2007: Hilbert spaces and operator algebras from categories

Suppose we have a category C of “configurations and processes” and an “action” functor S : C → (ℝ, +) giving the phase (set ℏ = 1 henceforth) e^{iS} : C → (U(1), ·) describing the amplitude for any process to occur.

How do we get a Hilbert space out of this? (One approach is to try and get a Cauchy surface; however, we will try something different.) Here’s one avenue of attack:

8.1. (Pre-)Hilbert spaces from categories. First, as a zeroth approximation to our Hilbert space, form a vector space as follows: let Ob(C) (resp. Mor(C)) be the set of all objecs (resp. morphisms) in C. Then we have s, t : Mor(C) → Ob(C), assigning to any morphism γ : x → y its source s(γ) = x and target t(γ) = y respectively.

Form the vector space Fun(Ob(C)) of “nice” complex-valued functions on Ob(C) - where we’ll have to see what “niceness” is required.

Then define for ψ, ϕ ∈ Fun(Ob(C)) an “inner product”:

⟨ϕ, ψ⟩ := ∫_{γ:x→y} e^{iS(γ)}\overline{ϕ(y)}ψ(x) 𝒟γ𝒟x𝒟y = ∫_{Mor(C)} e^{iS(γ)}\overline{ϕ(t(γ))}ψ(s(γ)) 𝒟γ

Remark 8.1.

(1) For this to make sense, we need a measure on Mor(C), and ψ, ϕ should be nice enough so that the integral converges - for instance, ψ ∘ s, ϕ ∘ t ∈ L^{2}(Mor(C)).

(2) Under “nice” conditions, given a measure 𝒟γ on Mor(C), we can find a measure on Ob(C) × Ob(C), and for any point (x, y) here, a measure on Mor_{C}(x, y), so that 𝒟γ ↔ 𝒟γ 𝒟x 𝒟y.

Now we have questions:

(1) Is ⟨ϕ, ψ⟩ linear in ψ and antilinear in ϕ?

(2) Is \overline{⟨ϕ, ψ⟩} = ⟨ψ, ϕ⟩?

(3) Is ⟨−, −⟩ nondegenerate? That is, given ϕ so that ⟨ϕ, ψ⟩ = 0 ∀ψ, is ϕ = 0?

(4) Is ⟨ψ, ψ⟩ ≥ 0 for all ψ?

Consider these in turn:

(1) is obvious if the integral is well-behaved.

(2) is more interesting:

(a) On the one hand,

\overline{⟨ϕ, ψ⟩} = ∫_{γ:x→y} e^{−iS(γ)}\overline{ψ(x)}ϕ(y) 𝒟y 𝒟x 𝒟y

whereas

⟨ψ, ϕ⟩ = ∫_{γ:x→y} e^{iS(γ)}\overline{ψ(x)}ϕ(y) 𝒟γ𝒟x𝒟y = ∫_{γ:y→x} e^{iS(γ)}\overline{ψ(y)}ϕ(x) 𝒟γ𝒟x𝒟y upon relabelling x ↔ y. Thus, the equality of the two comes from “time reversal symmetry”. It’s easy if C is a groupoid, since then, given γ : x → y, we get γ^{−1} : y → x - and since S is a functor, S(γ^{−1}) = −S(γ). So we’ll get \overline{⟨ϕ, ψ⟩} = ⟨ψ, ϕ⟩ if the measure 𝒟γ on Mor(C) is preserved by the transformation −1 : Mor(C) → Mor(C).

(b) But - our favourite example is not a groupoid! Recall - given a manifold Q, we have a category with Ob(C) = ℝ × Q, and a morphism γ : (t_{1}, q_{1}) → (t_{2}, q_{2}) is a path γ : [t_{1}, t_{2}] → Q with γ(t_{i}) = q_{i}.

[figure: path from (t_{1}, q_{1}) to (t_{2}, q_{2})]

Here, we’ve been assuming t_{1} ≤ t_{2}, so this is not a groupoid. We could adjoin inverses to get a groupoid, but then we’d get morphisms like

[figure: two paths between (t_{1}, q_{1}) → (t_{2}, q_{2}) → (t_{1}, q_{3}) - so there’s backwards motion in time!]

Such morphismsm do indeed show up in Feynman diagrams involving antimatter, but would require further thoughts.

(c) Research topics:

(i) Study Feynman’s original work on path integrals for a special-relativistic particle, and see if he allowed paths like

[figure: path that has parts moving both ways in time]

(ii) If so, formalize what he did using some category C. Is it

a groupoid, or merely a ∗-category?

(d) A ∗-category is a category C with a contravariant functor ∗ : C → C that is the identity on objects, and satisfies ∗∗ = 1_{C}.

Equivalently, for any morphism γ : x → y, there is a morphism γ^{∗} : y → x so that

(i) (γ_{1} ∘ γ_{2})^{∗} = γ_{2}^{∗} ∘ γ_{1}^{∗}

(ii) (γ^{∗})^{∗} = γ (These imply that (1_{x})^{∗} = 1_{x} ∀x ∈ Ob(C).) This is also called a category with involution, an involutive category, or in quantum computing, a †-category.

(e) We now make some remarks.

Remark 8.2.

(i) An obvious eample is a groupoid with one object - also called a group! Then ∗ is the inverse map on morphisms. (ii) The main example, however, is the category of Hilbert spaces and bounded linear operators, denoted by Hilb: given T : H → H′, we get the adjoint operator T^{∗} : H′ → H, defined by

⟨T^{∗}ϕ, ψ⟩ := ⟨ϕ, T ψ⟩

(iii) The requirement that x^{∗} = x for all objects x of C is somewhat “evil”; we might ask for a “non-strict” (ala monoidal categories) version only.

(3) ⟨−, −⟩ is usually degenerate, but that’s not bad; we can form the vector subspace K ⊂ Fun(Ob(C)) by

K = {ψ : ⟨ϕ, ψ⟩ = 0 ∀ϕ ∈ Fun(Ob(C))}

and form the quotient space

H_{0} = Fun(Ob(C))/K

on which we have ⟨−, −⟩ defined by

⟨[ϕ], [ψ]⟩ := ⟨ϕ, ψ⟩

Then this new ⟨−, −⟩ is nondegenerate on H_{0}.

(4) Is ⟨ψ, ψ⟩ ≥ 0?

To get this, we need some extra conditions - but we’d need to look at some examples to find nice sufficient conditions. This is somehow related to “reflection positivity” in the Osterwalder-Schrader Theorem.

Anyhow, if we get that these properties all hold, then H_{0} is called a pre-Hilbert space; we can then complete it to get a Hilbert space.

8.2. Operators and multiplying them. Besides the issue of producing a Hilbert spaces, there’s the issue of operators. How can we get some “nice” operators in Fun(Ob(C))?

We can get them from elements F ∈ Fun(Mor(C)), some space of “nice” complex-valued functions on Mor(C):

(F ψ)(y) := ∫_{γ:x→y} F (γ)ψ(x) 𝒟γ 𝒟x

where “nice” means this converges.

In fact, we get an algebra of such operators with some luck:

GF (γ) = ∫_{𝒫} G(γ_{2})F (γ_{1}) 𝒟p

where we integrate over the set

𝒫 := {(γ_{1}, γ_{2}) ∈ Mor(C) × Mor(C) : γ_{2} ∘ γ_{1} = γ}

and 𝒟p is a measure on 𝒫.

Remark 8.3.

(1) This is “convolution”; Fun(Mor(C)) is called the “category algebra” of C.

(2) If we’re working over a groupoid C, the above integral can be converted to an integral only over morphisms to one point, by a “change of variables”; moreover, the measure here is one that has already shown up earlier above.

9. Mar 13, 2007: The big picture

9.1. The case of finite categories. Last time, we sketched how to get a Hilbert space from a category C (of “configurations” and “processes”) equipped with an “amplitude” functor

A : C → U(1)

There are lots of subtleties involving analysis, but these evaporate when C is finite (so all morphism spaces are finite sets, all integrals are finite sums). Then we form the vector space Fun(Ob(C)) - which now means all functions ψ : Ob(C) → ℂ.

Fun(Ob(C)) is isomorphic to ℂ[Ob(C)] - the space of formal linear combinations of objects of ℂ. (This corresponds to allowing superpositions in quantum mechanics.)

Then we define a ℂ-sesquilinear map

⟨−, −⟩ : ℂ[Ob(C)] × ℂ[Ob(C)] → ℂ

by

⟨y, x⟩ := ∑_{γ:x→y} A(γ) ∀x, y ∈ Ob(C)

Recall that A = e^{iS/ℏ} here. (The prefix “sesqui” means “one-and-a-half”, and this is particularly appropriate, since it is antilinear - hence only ℝ-linear - in the first coordinate, but ℂ-linear in the second coordinate.)

We’re doing a “path integral”, but now it’s a sum over morphisms - we’re implicitly using counting measure on Hom_{C}(x, y).

Take ℂ[Ob(C)] and mod out by

{ψ ∈ ℂ[Ob(C)] : ⟨ψ, ϕ⟩ = 0 ∀ϕ ∈ ℂ[Ob(C)]}

to get a vector space H with a nondegenerate sesquilinear form on it; if that’s positive definite, then H is a (finite-dimensional) Hilbert space.

9.2. Example: particle on a line. Alex Hoffnung and I (=JB) have been looking at examples like this:

[figure: “a particle on a line” : lattice in T vs. X plot]

In this example, • Ob(C) = {1, 2, …, T } × {1, 2, …, X}.

• Morphisms in C are freely generated (under the composition / concatenation operation, and including the identity) by morphisms γ : (t, x) → (t+1, y) for all t ∈ {1, 2, …, T −1} and x, y ∈ {1, 2, …, X}.

So a typical morphism in C is like

[figure: a zigzagging “rising” path in the lattice, connected but not necessarily from the bottom to the top]

Now C is a “quiver”.

If you choose the amplitude A : C → U(1) to be a discretized version of the amplitude for a particle on a line, we recover standard physics in the continuous limit.

9.3. From particles to strings. We really want to categorify all this. We now make a table as we have done earlier; we shall make explanatory remarks later on.

Particles  Strings

(1) A category C.

(1) A 2-category (or double category) C.

[figure: path γ : x → y] [figure: γ_{1}, γ_{2} : x → y and the enclosed area Σ - or a “rectangle” with horizontal edges γ_{i} and enclosed area Σ - the worldsheet of a string]

(2) A functor A : C → U(1) ⊂ ℂ. Here, U(1) is the 1-category with one object ∗, and the morphisms are the elements of U(1).

(2) A 2-functor : C → U(1)[1]. (This is explained later.)

(3) From A : C → U(1), we try to build a Hilbert space - but first we form the vector space Fun(Ob(C)), which, if C is finite, is just Hom(Ob(C), ℂ) ≅ ℂ[Ob(C)] (3) From A : C → U(1)[1] − Tor ≅ U(1)[1], we try to build a 2-Hilbert space F UN(OB(C)), which, if C is finite, is just Hom(OB(C), Vect_{ℂ}) ≅ (we hope) Vect_{ℂ}[OB(C)].

Here, OB(C) could be the category formed by discarding the 2-morphisms in our 2-category C - but this only works if C is strict. What to do in general? Good questions.

(4) We define ⟨−, −⟩ on ℂ[Ob(C)] by

⟨y, x⟩ = ∑_{γ:x→y} A(γ).

Here, we use U(1) ↪ ℂ to add elements of U(1) and get elements of ℂ.

(4) ⟨−, −⟩ on Vect_{ℂ} should satisfy

⟨y, x⟩ = ⨁_{γ:x→y} A(γ) ∈ Vect_{ℂ}.

Here we use U(1) − Tor ↪ Vect_{ℂ}; this is explained below.

Remark 9.1. We present some facts about torsors below; let us first address the other points in the table above.

(1) For any abelian group A and n ≥ 0, we can form an n-category A[n]: the objects form a singleton set {∗}, the 1-morphisms are {1_{∗}}, the 2-morphisms are {1_{1_{∗}}}, and so on, until the n-morphisms (between the unique (n − 1)-morphism and itself). This set is different - and equals A.

(2) While defining ⟨−, −⟩ in the case of strings, we use that U(1) − Tor ↪ Vect_{ℂ}, sending a torsor to its corresponding 1-dimensional vector space (that contains the circle U(1)). (In fact, U(1) − Tor ↪ ℂ − Tor?)

[figure: circle maps to shaded plane containing circle]

Moreover, this vector space is actually a Hilbert space. For more on this, see Daniel Freed’s Higher algebraic structures and quantization.

9.4. Digression on torsors. Given any group G, a G-torsor is a G-set isomorphic to G (viewed as a G-set via left-multiplication). Denote the set of G-torsors by G − Tor. In other words, “a torsor is a group that has forgotten its identity”.

Thus, G-torsor morphisms are G-set maps that are also bijections.

(1) If G is abelian, then G − Tor is a monoidal category with

X ⊗ Y := X × Y/{(xg = gx, y) ∼ (x, gy)}

where X, Y ∈ G − Tor and g ∈ G acts on the right on X (since G is abelian and acts on X).

(2) If G is abelian, then G − Tor is a 2-category as follows: it has one object ∗; G-torsors as morphisms, composed using ⊗ defined above; and G-torsor morphisms as 2-morphisms.

(3) Next, recall the definition of the 2-category G[1] for the abelian group G. We now have that G[1] is a skeleton of G − Tor - so we have G[1] ≅ G − Tor.

To see this, we need one “special” morphism for every object. Thus, identify one of the torsors with G! This is to correspond to the identity morphism, since we can verify that G ⊗ G = G (where ⊗ was defined above).