Archive for the ‘Carlip's Class’ Category

Dr. Carlip’s Course on Quantum Gravity (pt 6)

15 May 2007

[Edit: This post is dedicated to Ed Clark, the fellow who first prodded me into physics; he survived the Hurricane Katrina, had a stroke, and I lost contact with him. He's in my thoughts to date.]

This basically continues the last lecture on quantization of constrained systems.

First Example: Electromagnetism

The electromagnetic tensor F_{\mu\nu}=A_{\nu ,\mu} - A_{\mu ,\nu} has a gauge invariance; the following are mathematical niceties

A_{\mu}\rightarrow A_{\mu} + \partial_{\mu}\Lambda

E^{i} = F^{i0}

B_{i} = \frac{1}{2}\epsilon_{ijk}F^{jk} .

L = \frac{1}{4} F_{\mu\nu}F^{\mu\nu} .

Exercises: Demonstrate the following:

1) The canonical variables are A_{i}, \pi^{i}=E^{i}, \{ A_{i}(x), E^{j}(x^{\prime}) \}_{t=constant} \sim \delta_{i}^{j} \delta^{3}(x-x^{\prime}) /

2) A^0 is a lagrange multiplier and the corresponding constraint is the Gauss Law \nabla\cdot E = 0.

3) The constraint generates gauge transformations:

\{ \int\Lambda (x) \nabla\cdot E(x) d^{3}x, A_{i} (x^{\prime}) \} = A_{i}(x^{\prime}) + \partial_{i}(x^{\prime})

\{ \int\Lambda (x) \nabla\cdot E(x) d^{3}x, E^{j} (x^{\prime}) \} = 0

Second Example: The Parametrized Nonrelativistic Particle

This second example is actually surprisingly relevant. [edit: I found this dissertation from 1995 by a one Mr. G. Ruffini, it's a good dissertation that's relevant here.] Recall the typical Lagrangian for a particle:

L = p_{i}\dot{q}^{i} - H(p,q)

The corresponding action is thus

I = \int L dt = \int (p_{i}\dot{q}^{i} - H)dt = \int (p_{i}dq^{i} - Hdt)

We introduce new variables q^{mu} = (t, q^{i}), a new parameter \tau which represents the time along the path of the particle (somewhat reminiscent of the proper time of relativity), p_0 = -H. We can rewrite the action as:

I = \int (p_{\mu}\frac{dq^{\mu}}{d\tau } - \lambda (p_0 + H))dt .

We have the gauge invariance: \tau\rightarrow\tau^{\prime} = \tau + \delta\tau

\delta p_{\mu} = \frac{dp_{\mu}}{d\tau }\delta\tau

\delta q^{\mu} = \frac{d q^{\mu}}{d\tau }\delta\tau

\delta\lambda = \frac{d}{d\tau } (\lambda\delta\tau )

Gauge invariance here is the same as parametrization invariance. The constraint is C = p_{0} + H.

\{ \epsilon (p_{0} + H), p_i \} = \epsilon\frac{\partial H}{\partial q^{i}} = -\epsilon\frac{dp_{i}}{dt} = (-\epsilon\frac{d\tau }{dt})\frac{dp_{i}}{d\tau }

\{ \epsilon (p_0 + H), q^i \} = -\epsilon\frac{\partial H}{\partial p_{i}} = -\epsilon\frac{dq^{i}}{dt} = (-\epsilon\frac{d\tau}{dt})\frac{dq^{i}}{d\tau}

\{ p_\mu , q^\nu \} = \delta_{\mu}^{\nu}

\{ \epsilon (p_0 + H), p_0 \} = 0 = \delta\tau\frac{dp_0 }{d\tau }

\{ \epsilon (p_0 + H), q^{0} \} = -\epsilon = \delta\tau\frac{dt}{d\tau } = \delta\tau\frac{dq^{0} }{d\tau }

Recall the picture of the gauge orbits and so forth. Well, there are two approaches to quantizing this picture:

1) Reduced Phase Space Quantization the motto “Constrain then quantize”

2) Dirac Quantization the motto “Quantize then Constrain”

There are some theorems from symplectic geometry that says that reduced phase space quantization is almost always possible.

Reduced phase space quantization of the parametrized particle is seemingly elementary:

I = \int (p_\mu \frac{dq^{\mu}}{d\tau} - \lambda (p_0 + H)) d\tau solve the constraint p_0 = -H .

I = \int (p_i \frac{dq^{i}}{d\tau} - H \frac{dt}{d\tau})d\tau gauge fix \tau = t.

I = \int (p_i\frac{dq^{i}}{dt} - H)dt which is the regular parametrized particle action that we began with.

In principle, Reduced phase space quantization is simple, in practice it is not always so simple. It can result in nonlocal conditions. Typically, gauge-fixing results in nonlocality. So the usual choice is Dirac quantization.

Example of Nonlocality in Electromagnetism

The gauge invariance is A^{\mu}\rightarrow A^{\mu }+\partial_{\mu }\Lambda . Choose the Lorenz gauge (not to be confused with Lorentz, as in Lorentz transformation in special relativity!): \partial_{\mu}A^{\mu} = 0 .

We can rewrite the potential in two terms:

A^{\mu} = \bar{A}^{\mu} + \partial^{\mu}\Lambda where \partial_\mu A^\mu = 0 .

The first term gives the line on the constrained surface where all the gauge orbits intersect the line once, and the second term involving lambda gives the gauge orbit. We can spot easily that

\partial_\mu A^\mu = \Box\Lambda

\Lambda = \Box^{-1}(\partial_{\beta}A^{\beta})

\bar{A}^{\mu} = A^\mu - \partial^{\mu}\Box^{-1}(\partial_{\rho}A^{\rho})

The second term (\partial^{\mu}\Box^{-1}(\partial_{\rho}A^{\rho})) is a nonlocal expression.

Back to the Parametrized Particle (Dirac Quantization)

Recall the action:

I = \int (p_\mu \frac{dq^{\mu}}{d\tau} - \lambda (p_0 + H))d\tau

NOTE: The Hamiltonian is effectively zero, the parametrized Hamiltonian is \lambda (p_0 + H)=0 when parametrized; this is to be expected as gauge invariance is translation in \tau.

Recall Unruh’s quantum clock that could possibly run backwards, it runs backwards if H_{eff} > 0 . Perhaps this parametrized particle has some deep meaning that avoids the problem. Note that for generally covariant systems H_{eff}=0 always!

Now we can introduce the wave function \Psi [q^{\mu}] , and define the momentum operator as \hat{p}_{\mu} = -i\hbar\frac{\partial}{\partial q^{\mu}} . There is an “auxillary Hilbert Space”, i.e. a Hilbert space not of physical degrees of freedom. We then impose the constraint
(p_{0} + H) \Psi [q^{\mu}] = 0 or equivalently

(-i\hbar\frac{\partial}{\partial t} + H)\Psi = 0 i.e. the Schrodinger equation is derived from this constraint.

We still have to define the inner product

\langle \Psi | \Phi \rangle_{aux} = \int d^{4}q \Psi^{*}\Phi is not a correct inner product for the schrodinger equation, there are too many degrees of freedom. We need to gauge fix the inner product, e.g. choose q^{0}=t, this gives

\int d^{3}q\Psi^{*}\Phi |_{q_{0}=t} which gives the correct inner product.

There is a beautiful paper by Woodard on this, Enforcing the Wheeler-DeWitt Constraints the Easy Way. Classical And Quantum Gravity 10 (1993) 483.

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Dr. Carlip’s Course on Quantum Gravity (pt 5)

14 May 2007

OK, this will be the last post for a few days, I have to write an essay on the rising use of double speak in advertisement in late capitalism (I know, the subject is just an irresistible one: advertisement = propaganda). However, this should not take more than a day and Dr. Carlip is in Canada for a black hole conference this Tuesday the 15th, so I may be able to cram a few more lectures in later tomorrow.

References:

Rivers, Path Integrals In Quantum Field Theory, Chapter 6.5

Mayes and Dowker, J Math Physics, 14 (1973) 434

DeWitt-Merette (I am not sure about the spelling on that second name) et al., Phys. Reports 50 (1979) 255

<The GRW model (I think this is a somewhat legitimate pdf) is a “deterministic”, that is non-wave function and additional stochastic mechanisms, interpretation.

Gerard ‘t Hooft is working on classical models that stochastically reproduce quantum models (his work actually looks fascinating, especially with automata!).> Note that I write asides in those sort of brackets, and now a little something about the cited works.

Mayes and Dowker specifically deal with operator ordering and the ambiguities that lie therein.

DeWitte-Morette deal with stochastic equations, measurements with path integrals, etc.

The path in path integrals are typically not nice and aren’t really differentiable.

Wiener measure is the mathematical formalism about changing coordinates in path integrals.

Quantizing Constrained Systems

Begin with a Lagrangian that depends on q and \dot{q} = dq/dt. Recall, if you haven’t seen this before, that the action is:

I = \int L(q, \dot{q}) dt

Varying this action gives the equations of motion: \ddot{q}=F(q,\dot{q}). There ,ay be terms with two time derivatives in the Lagrangian which can be “converted” into terms with one time derivative via integration by parts.

There is a general theorem by Ostrogradski that says in general Lagrangians with greater than one time derivative gives unstable systems, in general where the Hamiltonian is not bounded below so there is no minimum energy. Here there is another reference Carlip gave us (the man is like a wizard with his ability to seemingly conjure citations from thin air): Woodard, astro-ph/0601672.

The easiest way to turn this into a constrained system is with a new action:

I^{\prime} = \int (L(q, \dot{q}) + \lambda C(q, \dot{q}) )dt

where \lambda = Lagrange Multiplier, then

\ddot{q}=F(q,\dot{q}), C(q,\dot{q})=0, \dot{\lambda}=G(q,\dot{q})

Just another note, the constraints in General Relativity is G^{t}_{\mu} = 0.

In the canonical form, the constrained action takes the form:

I^{\prime} = \int (p\dot{q} - H + \lambda^{\alpha}C_{\alpha}(p,q))dt

First class constraints will be covered first (there are two classes of constraints: first and second; second class constraints are constraints which are not first class constraints). To have an intuitive picture of what constraints mean, it’s a surface in the phase space. One use of constraints is suppose we have N particles with a constant total energy and momentum, that would create a 4N dimensional surface in the phase space where the state of the system must be (supposing it were a relativistic system). Now, back to first class constraints, supposing we have several constraints:

[ \hat{C}_{1}, \hat{C}_{2} ] = 0 \Rightarrow \hat{C}_{1} \left| \psi \right> = 0 and \hat{C}_{2} \left| \psi \right> = 0 both cannot hold!

There are two conditions for a first class constraint (according to these notes of mine, note that I was rushing to jot all this down; Dr. Carlip appeared to be rather excited to teach this):

(1) \{ C_{\alpha}, C_{\beta} \} = f_{\alpha\beta}^{\gamma}C_{\gamma}

(2) \{ C_{\alpha}, H \} = V_{\alpha}^{\beta}C_{\beta}

These two conditions must be fulfilled for the constraint to be first class (first class constraints are easier than second class constraints in quantizing…I know from a nightmarish experience that sucked time away from my Winter Break; it was actually quite fun). If one or neither of these conditions are fulfilled, then you have second class constraints.

(First class) Constraints generate gauge-transformations, first class constrained systems are gauge invariant. Gauge choice is a coordinate choice on the surface in the phase space, I’ll cover this a little bit later on.

See the infamous Henneaux text for more information on quantization of gauge systems specifically, section 3.2.

Now we can find the variations of certain variables:

\delta q = \{ \epsilon C, q \}

\delta p = \{ \epsilon C, p \}

\delta\lambda = to be determined.

There is a wee bit of interesting math that I have nervously scribbled down in my notebook which I’ll reproduce here (if it’s wrong…and it’s most likely wrong…feel free to inform me):

\delta (p \dot{q} ) = \{ \epsilon C, p \} \dot{q} + p \frac{d}{dt} \{ \epsilon C, q \}

= \epsilon \dot{q} \frac{d C}{d q}+p \frac{d}{dt} ( -\epsilon \frac{d C}{d p} )

= -\frac{d}{dt} (\epsilon p\frac{\partial C}{\partial p} ) + \epsilon\dot{p}\frac{\partial C}{\partial p} + \epsilon\dot{q}\frac{\partial C}{\partial q}

= -\frac{d}{dt} (\epsilon p\frac{\partial C}{\partial p} ) + \epsilon\frac{d C}{d t}

= \frac{d}{dt} (\epsilon C - \epsilon p \frac{\partial C}{\partial p}) - \dot{\epsilon}C

And \delta H = \{ \epsilon C, H \} = \epsilon V C with one constraint:

\delta C = \{ \epsilon C, C \} = 0. Now we can find the variation of the action:

\delta I^{\prime} = (boundary terms) + \int ( -\dot{\epsilon} C - \epsilon V C + (\delta\lambda )C) if \delta\lambda = \dot{\epsilon} + \epsilon V , then \delta I^{\prime} = 0 .

In the phase space, there is a surface called the constrained surface (a fibre bundle - yeah I randomly spell words like a British fellow). We will continue talking about quantization of constrained systems next post too, and perhaps if I can doodle with GNUpaint well enough I might upload an image of the constrained surface in the phase space.

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Dr. Carlip’s Course on Quantum Gravity (pt 4)

9 May 2007

OK, he started by giving us four citations:

Isham, Relativity, Groups, and Topology II (1983)

Gotay, dg-ga/9605001

Tillman, gr-qc/0610141

Landsman, quant-ph/0506082

The general topic continues to be how to quantize classical systems. Isham deals with quantization in general, the cited text is actually lecture notes from summer school (Carlip thought very highly of them). Gotay deals with obstructions with quantization, problems of simply taking the Poisson bracket and turning them into commutators. Tillman’s paper is an introduction to deformation quantization while Landsman’s paper is on different approaches to quantization.

Recall from last time that we’d like quantization to change classical poisson brackets into commutators of operators on some hilbert space

\{ A, B \} \rightarrow \frac{-i}{\hbar} [\hat{A}, \hat{B} ]

Groenewold and von Hove showed that there is no quantization map from phase space to Hilbert space, etc. There is no ordering convention which is the real problem.

In ordinary quantum mechanics, you are never told this problem. You’re told to do it with just position and momentum, which is a choice of coordinates in the phase space!

\{ q, p \} \rightarrow \frac{-i}{\hbar} [\hat{q}, \hat{p} ]

There was a group of physicists in the early 1900s that argued that the action-angle coordinates was preferred instead for quantization (see section I of this (pdf)), but it’s empirically wrong since they (historically anyways) didn’t predict planck’s constant.

In ordinary quantum mechanics, recall that the operators e^{ia\hat{q}} and e^{ib\hat{p}} generate translations for position and momentum respectively. (Oh, a brief aside on what to do with e to the power of an operator, you change it into the series using the formal definition). In other words:

e^{ia\hat{q}/\hbar}\hat{p}e^{-ia\hat{q}/\hbar} = \hat{p} + a

We usually write everything in terms of \hat{q} and \hat{p} in quantum mechanics. These translation operators are transitive symmetry groups, they are also generators for the group. You can generalize this for curved phase spaces, use the translators to generate the group. Sometimes you can get 2 or more groups acting on the phase space, other times you don’t get a good answer and you have to guess.

Quantum mechanics on a half-line.

Suppose we were working in one dimension, and had a particle that could move on way on a line starting from the origin (for simplicity, it moves in the positive x direction). Take the basic variables to be x and xp = D.

e^{iaD/\hbar}xe^{-iaD/\hbar} = e^{a}x

Notice that here the shift is multiplied rather than added (which would mean, regardless what value a is, the particle always moves forward). The commutation relations are now:

[ x, D ] = i\hbar x (affine commutators)

(Note that D = 1 dimensional conformal transformation.) You still have to start with the classical Hamiltonian, and be careful about the operator ordering to make the Hamiltonian Hermitian. A fellow by the name of John Klauder has been working on affine commutation relations and coherent state path integrals.

This is a fundamental issue of quantization and it is not clear whether or not it is fundamental enough.

One way to approach this in quantum gravity is by making the spatial metric:

g_{ij} = e^{\phi_{ij}}

using \phi_{ij} instead. No one figured out how to get very far with this.

The other way to go is by making the metric a fuction of 2 tetrads:

g_{ij} = e^{a}_{i}e_{ja}

This is the starting point for loop quantum gravity.

There are a couple of other approaches to quantization.

One rather new (and exciting!) approach is deformation quantization. It has been worked on largely by mathematicians. The idea is that if you can’t directly get the commutator, you instead do the following:

\{ A, B \}\Rightarrow\frac{-i}{\hbar} [ \hat{A}, \hat{B} ] + O(1) + O(\hbar ) + ...

Where O(1) + O(\hbar ) + ... are corrections to the commutator. Uniqueness is a little fuzzy too but it’s unique up until unitary transformations. (If the graviton had a half-integer spin, we’d need to use anti-commutators.)

The map from a phase space function to an operator on a Hilbert space:

A\rightarrow Q(A)

\lim_{\hbar\to 0} || \frac{-i}{\hbar} [ Q(f), Q(g) ] - Q(\{ f, g\} ) || = 0 (I have scribbled below the limit an arrow pointing to it saying “Might be weak topological norm”)

You can define a classical product called the star product:
A*B = Q^{-1}(Q(A)Q(B)) \neq B*A

Defrmtion quantization really means changing the product to a noncommutative product:

f*g = f exp\frac{i\hbar}{2}(\frac{\overleftarrow{\partial}}{\partial x^{\mu}}\frac{\overrightarrow{\partial}}{\partial p_{\mu}} - \frac{\overrightarrow{\partial}}{\partial x^{\mu}}\frac{\overleftarrow{\partial}}{\partial p_{\mu}}) g

You, of course, expand this out as an infinite series:

f*g = \sum_{n=0}\frac{1}{n!} (\frac{i\hbar}{2})^{n} (\frac{\partial^{n}f}{\partial (x^{\mu})^{n}}\frac{\partial^{n}g}{\partial (p_{\mu})^{n}} - \frac{\partial^{n}g}{\partial (x^{\mu})^{n}}\frac{\partial^{n}f}{\partial (p_{\mu})^{n}})

But we simplify this to be:

f*g = fg + \frac{i\hbar}{2}(\frac{\partial f}{\partial x^{\mu}}\frac{\partial g}{\partial p_{\mu}} - \frac{\partial g}{\partial x^{\mu}}\frac{\partial f}{\partial p_{\mu}}) + O(\hbar^{2})

Thus we can find:

f*g - g*f = i\hbar\{ f, g \} + O(\hbar^{2})

where O(\hbar^{2}) is corrections of order \hbar^{2} (i.e. very small, of order 10^{-66} in mks units).

Perhaps there is some important work from noncommutative geometry that could be carried over to deformation quantization, and vice-versa(?). Note that order \hbar corrections can be very important to quantum gravity.

There is one other approach completely different from canonical quantization, and that is the Feynman path integral quantization. For all practical purposes, the path integral quantization is done on a lattice, and we change derivatives to:

\frac{dx}{dt}\rightarrow \frac{x(t+\Delta t)-x(t)}{\Delta t}

and integrals to

\int f(x) d^{4}x = a^{4} \sum_{i} f(i)

where a is the lattice spacing. Path integration, however, does not escape the ambiguities of quantum gravity…it only hides it quite nicely.

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Dr. Carlip’s course on quantum gravity (Pt 3)

9 May 2007

Why haven’t we quantized gravity yet?

The first attempt that Carlip was aware of was in 1930 by Rosenfield. There have been at least 10 nobel prize winners working on quantum gravity(!).

Actually, effective action and effective potential were invented by Bryce DeWitt as an attempt to quantize gravity.

The two best attempts today are Loop Quantum Gravity and String theory. There is also development in dynamical triangular lattices which shows promise.

The basic gauge we’ll be dealing with is gauge invariance, i.e. coordinate-frame independent invariance.

It’s very difficult to talk about the coordinates in curved spacetime. We speak of the amount of time it takes for radiation to hit mercury and return according to a nuclear clock, instead of the position of mercury.

Worse, all the variables of general relativity are nonlocal (which isn’t necessarily a bad thing! Nonlocality is like cholesterol: there is the bad kind and the good kind, this is the good kind). Quantum Field theory deals with local variables. The infamous “problem of time” is an example of this inconsistency.

The role of time in quantum mechanics is used for renormalization and time-evolution. Because there are no preferred time coordinates, this complicates things. In quantum gravity, it is really difficult as the time coordinate could be superpositioned and then overly complicated.

Torre and Varadarajan (I think it is “Functional evolution of free quantum fields,” Class. quantum grav., 16, 2651 (1999), I don’t have the citation written down) had a thought experiment: take a flat spacetime and take an initial time t=0 and a final time t=1. They go through and show essentially, at minimum, that the Schrodinger picture and the Heisenberg picture are inequivalent. Carlip thinks that we ought to use the Heisenberg picture. Mind you however this is for flat spacetime. Perhaps quantum gravity could demonstrate the two pictures really are equivalent?

There are three parts to this problem of time:

1) We don’t know how to renormalize wave functions, and thus cannot really know how to get probabilities

2) We don’t know if quantum gravity is generally covariant

3) The Hamiltonian is 0 in a Hamiltonian formulation of classical general relativity.

The third problem means that we don’t know the right type of time evolution. Perhaps we could extract information about the time from the spatial metric? Still an open problem today.

The Next Problem: Causality

In ordinary flat spacetime you have the light cone. In ordinary quantum field theory, causality is “built-in” with the assumption that two operators seperated by a spacelike curve will commute. This is “microcausality”.

In quantum gravity, you don’t have fixed coordinates, and thus you don’t have fixed light cones. We don’t know how serious this is since quantum gravity has nonlocal operators. That could save or sink us.

Another Remark on Time

You measure time with a clock. You can’t have a clock with no matter and no energy, which means you are not asking the same question you began with. That’s not such a serious problem as you could use the proper time of world lines as a clock.

Also if you measure time with a clock made out of quantum matter, there is a finite positive probability that the clock will run backwards.

Problem with Vacuum

There may be a minimum measurable distance. If that’s true, our picture of a smooth spacetime breaks down and we have nothing to replace it with. Wheeler thought that, since virtual particles are created and annihilated at small time intervals, there would be a sort of space foam. There is no clear way how to get from there to quantum gravity however.

How to Solve These Problems

The way to solve these problems is not to stare at them and wish them away, but to look at current paradigms, how they resolve these problems, and try to use similar ideas from them.

What does it mean to “quantize” something anyways?

That is, physically speaking, the wrong question. Presumably, the world all ready is quantized and the classical picture is only an approximation. In practice, however, this does not stop us from quantizing classical systems.

The test for the classical limit is usually the weak field approximation. There are strong field approximations for quantum gravity involving pulsars and so forth.

What does it mean to quantize a system? Well, in 1 dimensional quantum mechanics, you start with the commutation relations

[ \hat{q}, \hat{p} ] = i\hbar

and find the Hilbert space where the relation works. Recall from classical mechanics, the Poisson bracket is:

\{ q, p \} = 1.

1) Start with the poisson brackets and change them to commutators:

\{ q, p \} \Rightarrow \frac{-i}{\hbar} [ \hat{q}, \hat{p} ]

Problem: We have to worry whether there is a unique Hilbert space for this system. There is a proof from von Neumann that there is a unique Hilbert space to define these commutators. This theorem does not hold for quantum field theory (it has an infinite number of dimensions as opposed to a finite number). As a matter of fact, there are many representations of the free Klein-Gordon scalar field.

Moral: Algebra of operators does not equate to the existence of a unique quantum theory.

Problem: This requires a choice of a q and a p, i.e. a coordinate choice! A change in coordinates in classical mechanics is a canonical transformation; there is a problem with this in quantum mechanics because of the operator ordering problem. There is a theorem about this by van Hove that shows it is impossible to do this: \{ q, p \} \Rightarrow \frac{-i}{\hbar} [ \hat{q}, \hat{p} ]

Quantization “Map”

We would like there to be a quantization map

Q: phase space observables \rightarrow operators

such that:

1) Q(c_{1}f + c_{2}g) = c_{1}Q(f) + c_{2}Q(g) i.e. it’s linear

2) Q(1)=1

3) Q(x) and Q(p) are replaced irreducibly, a technical equation to get back the ordinary schrodinger equation.

4) Q( \{ A,B\} ) = \frac{-i}{\hbar} [ Q(A), Q(B) ]

You might want to weaken the third condition. Von Hove’s theorem basically says that there is no such map Q because of the operator ordering problem, e.g.

x^{2}p^{2} = \frac{1}{9} \{ x^{3}, p^{3} \} = \frac{1}{3} \{ x^{2}p, p^{2}x \}

However according to our map we get:

\frac{1}{9} [ \hat{x}^{3}, \hat{p}^{3} ] = \frac{1}{3} [ \hat{x}^{2}\hat{p}, \hat{p}^{2}\hat{x} ] + \frac{4}{3}\hbar^{2}\hat{x}\hat{p} - \frac{2}{3} i\hbar^{3}.

Which clearly is not \frac{1}{3} [ \hat{x}^{2}\hat{p}, \hat{p}^{2}\hat{x} ]!

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Dr. Carlip’s course on quantum gravity (Pt 2)

8 May 2007

OK, so I’ll basically recount the lectures here as best as I can remember. The first lecture dealt with the Eppley and Hannah thought experiment (their work: Eppley and Hannah, Foundations of Physics 7 (1977) p.51), it deals with the uncertainty principle in classical spacetime. (Recall the Heisenberg Microscope) Basically, if General Relativity is classical and matter is quantized, then we can violate the uncertainty principle.

Carlip commented that perhaps it’s possible that gravity saves you from the wave function collapse, and thought about re-analyzing the Eppley and Hannah thought experiment with the limits of classical gravitational waves in mind.

Carlip went on to mention, to my disgust, that Quantum Cosmologists prefer the Many-Worlds Interpretation of Quantum Mechanics. I dislike it because it has this “wave function of the universe” which seems cartoonish; Rovelli’s paper on Relational Quantum Mechanics actually suggests that one could get from the Many-Worlds interpretation to the relational interpretation by completely disregarding the wave function of the universe as meaningless (if I recall correctly).

We went on to discuss semiclassical gravity, where General Relativity is left as classical but the matter and energy is treated as quantum. He gave us another citation for this section (Page and Geilker, Physics Review Letters 47 (1981) p. 979. This changes Einstein’s field equation to
G_{\mu\nu} = 8\pi \hat{T}_{\mu\nu}
For the Einstein curvature tensor of spacetime G_{\mu\nu}, and the energy-momentum operator \hat{T}_{\mu\nu}. This does not make sense since we are equating a tensor to an operator (it’s like equating a matrix of numbers to a matrix of operators, doesn’t work!). So we could possible make it an Eigenvalue equation:
G_{\mu\nu} \left| \psi\right> = 8\pi\hat{T}_{\mu\nu} \left| \psi\right>
Another possibility is to have
\langle \psi | G_{\mu\nu} | \psi\rangle = 8\pi \langle \psi | \hat{T}_{\mu\nu} | \psi\rangle
which is the expectation value of the quantized stress-energy tensor is equal to the “expectation value” of the Einstein tensor. This approach is called semi-classical gravity and was seriously considered in the 1960s.

First thing to notice is that semi-classical gravity makes quantum mechanics nonlinear. This nonlinearity is “very small”, perhaps if we used Newtonian gravity:
(Schrodinger equation) i \hbar\frac{\partial}{\partial t}\psi = \frac{- \hbar^{2}}{2 m}\nabla^{2}\psi + mV_{grav}\psi
(Newtonian field equation) \nabla^{2} V_{grav} = 4\pi G \langle \psi | \rho | \psi \rangle
where \rho is the density of the particle, G is the Newtonian gravitational constant. The nonlinearity comes from the fact that \psi depends on V which in turn depends on \psi again. This is the Schrodinger-Newton equation. This effect may be testable in the next 10 years; Carlip gave us another reference (http://arxiv.org/gr-qc/0606120 is the preprint of the paper Carlip and one of his students worked on).

There are some serious problems with this approach. For example, the covariant derivative of the Einstein tensor is supposed to be 0 by the “bianchi identities”:
\nabla_{\mu}G^{\mu\nu}=0
but for the “quantum” stress-energy tensor we get:
\nabla_{\mu} \langle \hat{T}^{\mu\nu} \rangle= \langle \nabla_{\mu} \hat{T}^{\mu\nu} \rangle \neq 0
and we have an inconsistency, by the conservation of energy we should get this second equation to be 0. That’s not too serious a problem, but it’s something to think about at the very least.

There are a variety of other complications, like field redefinitions. It’s ambiguous enough in quantum field theory, but now it’s extraordinarily ambiguous! This is not a “fatal” objection, but it introduces complications.

Moral of the story: Gravity needs to be quantized. Though none of this is conclusive proof.

There are some positive arguments why quantum gravity is good. In particular it would be nice if there was a model without an initial singularity. Maybe if we quantize gravity, we will lose the singularity.

Maybe quantum gravity will cure the divergences of quantum field theory. Look at the mass of a particle at some distance \epsilon:
m(\epsilon ) \approx m_{0} + \frac{e^2}{\epsilon} + ...
If you stick in gravity (classically):
m(\epsilon ) \approx m_{0} + \frac{e^2}{\epsilon} - G\frac{m^{2}(\epsilon )}{\epsilon } + ...
Perhaps the gravitational field potential could cut off higher terms.

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Dr. Carlip’s course on quantum gravity (Pt 1)

7 May 2007

Boy what a fun course! Dr. Carlip is a very funny man.

It has been a very enlightening experience so far. I have a new perspective on nonlocal operators; before I used to be influenced by the Relativist’s motto “Locality, Good! Nonlocality, evil!” But when you think about it with an example, say we want to know the position of Mercury. That’s not too well defined, what one really means is “The time it takes for radiation to hit mercury and return according to a nuclear clock”. It turns out that all the variables of general relativity are nonlocal…which makes it difficult to deal with quantization of the thing because quantization deals with local operators!

I suppose one would also have to take into account the effects of quantum gravity into quantum field theory. I mean, there would be a discrete (possibly finite?) number of points in space time. That would change the number of dimensions of the Hilbert space we would be dealing with! You see, originally quantum field theory had an infinite number of dimensions to represent classical, continuous spacetime! But if gravity is indeed quantized, it would no longer be a continuous spacetime! This would not change anything if it were still infinite however…it would just make it countably many, albeit infinite, dimensions of the Hilbert space.

Perhaps, if I may take up my role as Devil’s advocate, both theories are wrong! General Relativity is “obviously” quantizeable based on the empirical results of semiclassical gravity experiments. But there is something unappealing about quantum theory…chiefly its sole defect that it makes no sense! That’s just a small problem though…

There is hope however that there is a kernel of truth in both theories, perhaps both are approximations to an underlying theory that is heavily observer dependent? Who knows! I’m certainly not going to pretend to!

I’ll continue ranting about how great Carlip’s class is later…I have class to go to now.

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