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

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.

This entry was posted on 14 May 2007 at 11:38 pm and is filed under Carlip's Class. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “Dr. Carlip’s Course on Quantum Gravity (pt 5)”

  1. Thomas Larsson Says:
    15 May 2007 at 12:20 am

    A few notes which may clarify things.

    Constraints are second class if the Poisson bracket matrix

    [C_a, C_b] = D_ab

    is invertible, otherwise first class. If they are second class, you can replace Poisson brackets by Dirac brackets,

    [F, G]^* = [F, G] - [F, C_a] D^ab [C_b, G],

    (D^ab inverse of D_ab) and simply solve the constraints.

    It is two second class constraint that cannot both annihilate a physical state, because C_a |psi> = C_b |psi> = 0 implies D_ab |psi> = 0, i.e. |psi> = 0 since D_ab is invertible.

    First class count twice, because you can typically make them second class by adding a gauge fixing condition.

    the infamous Henneaux text

    You mean Henneaux and Teitelboim, no? Or Henneaux’ Phys Rep from the 1980s?

  2. angryphysicist Says:
    15 May 2007 at 1:05 pm

    Thanks for that Thomas, that really elaborated quite a bit :)

    And the infamous Henneaux and Teitelboim text, yes, the phonebook of quantizing gauge systems.

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