The Wheeler-de Witt equation
John Baez
>Assuming a compact universe, a problem concerning the Hamiltonian
>constraint arises. It looks as if only a single state (the Hartle-
>Hawking state, supposedly) satisfies this constraint, about of the same
>reasons as the reasons for a unique vacuum in a usual quantum field
>theory (am I mistaken?).
Yes, you're mistaken. There is a huge difference between the
Hamiltonian constraint in quantum gravity and the Hamiltonian
in ordinary quantum field theory. This is one of the main reasons
why quantum gravity is a tricky subject! It's not like the other
quantum field theories you know and love.
I meant to explain this in reply to a post by Jacques Mallah,
where he mused:
>if there's really just an energy eigenstate, there is no time and
>no dynamics.
The best way to start understanding this is to think about
*classical* general relativity - not *quantum* gravity.
If you start with the Lagrangian, turn the crank and work out
the Hamiltonian density for general relativity, you get a
quantity H which is zero when the equations of motion hold
(i.e. Einstein's equations). This is why we call it a
Hamiltonian "constraint" - the equations of motion imply the
constraint
H = 0
It follows that in general relativity there is not just *one*
solution of this equation - *every* spacetime satisfying Einstein's
equations gives a solution of this equation! It also follows
that there is real honest-to-goodness dynamics lurking in the
solutions to this equation.
We don't know for sure, but we sure expect that something similar
holds in quantum gravity. When we quantize the above equation we
expect to get the Wheeler-DeWitt equation
H psi = 0
where now H is an operator on some Hilbert space. And we expect
*lots* of solutions of this equation. And we expect real honest-
to-goodness dynamics to be lurking in the solutions to this equation.
It's just a bit hard to dig it out. For that, we need to "thaw
the frozen formalism" - something I don't have time to get into
now.
The idea of the Hartle-Hawking path integral is that it's supposed
to give one *particular* solution of
H psi = 0,
namely the solution that actually corresponds to our universe.
Now, there are lot of technical issues I am deliberately glossing
over here. I am just trying to give a zeroth-order approximation
to the full story (which is still a story in progress). The really
important point is that general relativity is truly revolutionary
compared to, say, Maxwell's equations. The whole framework is
different - once the geometry of spacetime itself becomes a dynamical
variable, the old stuff we know and love about Hamiltonians no longer
applies! Instead, we've got a Hamiltonian constraint... so quantum
gravity is drastically different from, say, quantum electrodynamics.
squ...@my-deja.com
ba...@galaxy.ucr.edu (John Baez) wrote:
> In article <8b7njb$3i3$1...@nnrp1.deja.com>, <squ...@my-deja.com> wrote:
>
> >Assuming a compact universe, a problem concerning the Hamiltonian
> >constraint arises. It looks as if only a single state (the Hartle-
> >Hawking state, supposedly) satisfies this constraint, about of the
> >same reasons as the reasons for a unique vacuum in a usual quantum
> >field theory (am I mistaken?).
>
> Yes, you're mistaken. There is a huge difference between the
> Hamiltonian constraint in quantum gravity and the Hamiltonian
> in ordinary quantum field theory. This is one of the main reasons
> why quantum gravity is a tricky subject! It's not like the other
> quantum field theories you know and love.
Interesting. I was mistakenly thinking the HH state is the only one the
constraint allows.
> The idea of the Hartle-Hawking path integral is that it's supposed
> to give one *particular* solution of
>
> H psi = 0,
>
> namely the solution that actually corresponds to our universe.
Ah, so its really just an "elegant" solution, no real reason to think
its the true one. Thx for correcting my ways! :-)
Regards, squark.
Charles Torre
> In article <8b7njb$3i3$1...@nnrp1.deja.com>, <squ...@my-deja.com> wrote:
>
>>Assuming a compact universe, a problem concerning the Hamiltonian
>>constraint arises. It looks as if only a single state (the Hartle-
>>Hawking state, supposedly) satisfies this constraint, about of the same
>>reasons as the reasons for a unique vacuum in a usual quantum field
>>theory (am I mistaken?).
>
> Yes, you're mistaken. There is a huge difference between the
> Hamiltonian constraint in quantum gravity and the Hamiltonian
> in ordinary quantum field theory. This is one of the main reasons
> why quantum gravity is a tricky subject! It's not like the other
> quantum field theories you know and love.
>
[I delete a nice discussion of the Hamiltonian constraint and its counterpart
in quantum gravity.]
Let me add (or maybe subtract) a little to this discussion
by suggesting an analogy.
In Schrodinger picture quantum mechanics the time evolution of
the wave function is such that
h psi = i d/dt psi ,
where Planck's constant is unity and h is the Hamiltonian.
To get a handle on what is happening with the Wheeler-De Witt
(WDW) equation, define
H = h - i d/dt,
so that the wave function satisfies
H psi = 0.
You can see that this is NOT a zero energy type of equation; it
is just the Schrodinger equation in notational disguise.
Now imagine that someone has mischievously made a coordinate
transformation that mixes up the spatial and temporal variables
in some hideous way. Now your equation H psi = 0 looks nothing
like the Schrodinger equation that it is, nevertheless,
equivalent to.
By the way, the quantity H is sometimes called a
"super-Hamiltonian".
One view on the Wheeler-DeWitt (WDW) equation is that it is
analogous to this final, complicated form of H psi = 0.
Canonical quantization of general relativity delivers dynamics
in the WDW form rather than in the Schrodinger form since there
can be no preferred time in a generally covariant theory.
In quantum gravity, just given the WDW equation, any
conventional quantum ideas in which the word "time" appears are
bound to be problematic since the theory doesn't give you the
dynamics relative to some time parameter. This is one facet of
"the problem of time". Wouldn't it be nice to uncover a hidden
time variable and cast the WDW into the more traditional
Schrodinger form? People have tried, but with very limited
success.
The analogy given above is, like all analogies, not perfect.
Still, you can get a good feel for the logical status of the WDW
equation from it.
Charles Torre
squ...@my-deja.com
to...@cc.usu.edu (Charles Torre) wrote:
> In Schrodinger picture quantum mechanics the time evolution of
> the wave function is such that
>
> h psi = i d/dt psi ,
>
> where Planck's constant is unity and h is the Hamiltonian.
>
> To get a handle on what is happening with the Wheeler-De Witt
> (WDW) equation, define
>
> H = h - i d/dt,
This is not an orperator on the usual quantum state space (d/dt is not).
The H of the Wheeler-de Witt equation is such an operator.
Regards, squark.
[A note for the moderator: please remove the deja thing.]
Sent via Deja.com http://www.deja.com/
Before you buy.
John Baez
>In article <0B9tNM...@cc.usu.edu>,
>to...@cc.usu.edu (Charles Torre) wrote:
>> In Schrodinger picture quantum mechanics the time evolution of
>> the wave function is such that
>>
>> h psi = i d/dt psi ,
>>
>> where Planck's constant is unity and h is the Hamiltonian.
>>
>> To get a handle on what is happening with the Wheeler-De Witt
>> (WDW) equation, define
>>
>> H = h - i d/dt,
>This is not an operator on the usual quantum state space (d/dt is not).
>The H of the Wheeler-De Witt equation is such an operator.
Actually Charles Torre's analogy here is quite good.
The H of the Wheeler-De Witt equation is not an operator on the
space of physical states of quantum gravity; it's an operator on a
bigger Hilbert space called the space of "kinematical" states. To
be a physical state, a kinematical state must satisfy the Wheeler-
De Witt equation
H psi = 0
together with some other constraints coming from other components
of Einstein's equation.
Similarly, we could start with a big space of "kinematical wavefunctions"
psi(t,x) - arbitrary functions on spacetime - and pick out the "physical
wavefunctions" by imposing the constraint
H psi = 0
where H = h - i d/dt as in Torre's post. While not explained in most
textbooks, this approach to quantum mechanics has some very nice features.
It puts time and space on more of an equal footing than the standard
approach, where wavefunctions are treated as functions on space from the
very start. Also, it helps one understand the issue of "thawing the
frozen formalism" and extracting dynamics from solutions of the Wheeler-
De Witt equation.
By the way, squark, I seem to spend a lot of time criticizing and
correcting stuff that you say, but this is just because you talk a
lot about things I'm interested in! I'm glad you decided to join
us here.
Also by the way, as a kind of "argumentum pro hominem", I might add
that Charles Torre is an ex-philosopher who saw the light and switched
to physics. He is a real expert on field theory, especially on issues
concerning symmetry, which are so important in quantum gravity - so
he's always worth listening to.
squ...@my-deja.com
ba...@galaxy.ucr.edu (John Baez) wrote:
> ...
> Similarly, we could start with a big space of "kinematical
> wavefunctions" psi(t,x) - arbitrary functions on spacetime - and pick
> out the "physical wavefunctions" by imposing the constraint
>
> H psi = 0
>
> where H = h - i d/dt as in Torre's post. While not explained in most
> textbooks, this approach to quantum mechanics has some very nice
> features. It puts time and space on more of an equal footing than the
> standard approach, where wavefunctions are treated as functions on
> space from the very start.
Fascinating! I thought of the idea, but never thought any real sense
can be put into it. Oh, but now anther problem arises - how does a one
go about this in quantum field theory? In particle physics, the
transition from the wavefunction psi(x) to the wavefunction psi(x,t) is
quite natural. In QFT, on the other hand, we have the wave functional
Psi[fi(x)]. What is his "natural extension"? Psi[fi(x),t] is
definitetly not natural, as it is reference frame dependent. It is
seducing to use Psi[fi(x,t)], but this doesn't work as the fields in
different points don't commute. So what happens?
> By the way, squark, I seem to spend a lot of time criticizing and
> correcting stuff that you say, but this is just because you talk a
> lot about things I'm interested in!
It's okey, I'm not offended. :-)
> Also by the way, as a kind of "argumentum pro hominem", I might add
> that Charles Torre is an ex-philosopher who saw the light and switched
> to physics. He is a real expert on field theory, especially on issues
> concerning symmetry, which are so important in quantum gravity - so
> he's always worth listening to.
I'll remember this.
Regards, squark.
Kwok Man Hui
John Baez wrote:
>
> In article <8bfpq0$j9f$1...@nnrp1.deja.com>, <squ...@my-deja.com> wrote:
>
> >In article <0B9tNM...@cc.usu.edu>,
> >to...@cc.usu.edu (Charles Torre) wrote:
>
> >> In Schrodinger picture quantum mechanics the time evolution of
> >> the wave function is such that
> >>
> >> h psi = i d/dt psi ,
> >>
> >> where Planck's constant is unity and h is the Hamiltonian.
> >>
> >> To get a handle on what is happening with the Wheeler-De Witt
> >> (WDW) equation, define
> >>
> >> H = h - i d/dt,
>
> >This is not an operator on the usual quantum state space (d/dt is not).
> >The H of the Wheeler-De Witt equation is such an operator.
>
> Actually Charles Torre's analogy here is quite good.
>
> The H of the Wheeler-De Witt equation is not an operator on the
> space of physical states of quantum gravity; it's an operator on a
> bigger Hilbert space called the space of "kinematical" states. To
> be a physical state, a kinematical state must satisfy the Wheeler-
> De Witt equation
>
> H psi = 0
>
> together with some other constraints coming from other components
> of Einstein's equation.
>
What about the "dynamical" states? The Schrodinger equation can describe
dynamics, so can the Wheeler-De Witt equation.
What about your spin foam? Does it satisfy any form of those constraint
equations?
K.M. Hui
squ...@my-deja.com
squ...@my-deja.com wrote:
> ...
> Fascinating! I thought of the idea, but never thought any real sense
> can be put into it. Oh, but now anther problem arises - how does a one
> go about this in quantum field theory? In particle physics, the
> transition from the wavefunction psi(x) to the wavefunction psi(x,t)
> is quite natural. In QFT, on the other hand, we have the wave
> functional Psi[fi(x)]. What is his "natural extension"? Psi[fi(x),t]
> is definitetly not natural, as it is reference frame dependent. It is
> seducing to use Psi[fi(x,t)], but this doesn't work as the fields in
> different points don't commute. So what happens?
Oh, I have thought of an answer myself. In the usual Heisenberg picture
for quantum field theory, the state is not labeled by anything, and the
field operators are space-time dependent. Equally, we can use a "super-
Shrodinger" picture, in which the quantum state is space-time
dependent, and the fields and their derivatives are fixed distinct
operators in the Hilbert space. Now, the analog of the Shrodinger
equation would be idPsi/dt=HPsi, idPsi/dx_k=-P_kPsi. If we lift this
requirement, we obtain the "generalization" I wanted. Now, we obtain 4
constraints instead of a one, but this shouldn't be surprising, as in
quantum gravity, we have both the Hamiltonian and the Diffeomorphism
constraints, so the situation is completely analogical. Comments and
replies are welcome.
P.S.
Note that apparent incovariance exists because Psi=Psi[fi(x)], but we
could decompose Psi with respect to a basis defined on any other Cauchy
hypersurface as well, so there is no real incovariance.
Best regards, squark.
John Baez
Kwok Man Hui <km...@purdue.edu> wrote:
>What about your spin foam? Does it satisfy any form of those constraint
>equations?
One can try to use spin foams to find solutions of the Wheeler-DeWitt
equations - Rovelli and Reisenberger wrote some papers on this. On
the other, it's equally plausible that the whole idea of trying to
find solutions of the Wheeler-DeWitt equation is ultimately misguided.
In loop quantum gravity, spatial diffeomorphisms don't have any
infinitesimal generator - so why should diffeomorphisms that push
time forward? These days, I've sort of given up on the Wheeler-DeWitt
equation. Maybe sometime later I'll decide this was overly pessimistic.
Who knows?
It's worth keeping in mind that the Wheeler-DeWitt equation is firmly
based on the notion that spacetime is a continuum. If spacetime is
something discrete, we'll probably need some discrete analog of
diffeomorphism-invariance, which will require some modification of
the Wheeler-DeWitt equation.
squ...@my-deja.com
ba...@galaxy.ucr.edu (John Baez) wrote:
> In loop quantum gravity, spatial diffeomorphisms don't have any
> infinitesimal generator - so why should diffeomorphisms that push
> time forward?
They don't? I thought there is something like a "loop space
derivative", can't it be used for the purpose?
Best regards, squark
km...@my-deja.com
> Kwok Man Hui <km...@purdue.edu> wrote:
>
> >What about your spin foam? Does it satisfy any form of those constraint
> >equations?
>
> One can try to use spin foams to find solutions of the Wheeler-DeWitt
> equations - Rovelli and Reisenberger wrote some papers on this.
Are you refering to their papers: Spin foams as Feynman diagrams
gr-qc/0002083, gr-qc/0002095 and some other papers quoted there like
"Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeous
space" hep-th/9907154 ? I've read partially gr-qc/0002095 (a connection
approach) because his paper has defect. Their work is to show spin foam
model can be derived as field theories over group manifolds (homogeneous
space). First, I don't have preprints about Barrett-Crane relativistic
model, so I don't know how Boulatov and Ooguri have generalized the matrix
models of 2d quantum gravity to 3d and 4d, and especially how they can answer
my question and in what form and how valid that form is, if we base on the
spirit of Ashtekar's new variables or its variations. Second, I don't have
background in that area. So I don't know how to judge their work with respect
to my questions, and any extension of that idea if they didn't show their
constraints algebra or habitat. Third, I don't remember which This Week's
Finds mentioned criticism about B-C model has trivial topological invariants
(I can't remeber what type of invariant). So I don't know how this triviality
(if it still exists) affect those lines of extension of B-O thought. Fourth,
field theories over group manifolds are too restrictive? At least we know
that most of our field theories are based on a metric not on group manifolds.
> In loop quantum gravity, spatial diffeomorphisms don't have any
> infinitesimal generator - so why should diffeomorphisms that push
> time forward?
I think this is related to "problem of time" in canonical quantum gravity.
I've read about 50 pages of Isham's paper Canonical Quantum Gravity and the
Problem of Time gr-qc/9210011. He has several interpretations of that awkful
equation. That may give us some ideas if we want to pursue weaker sense of
"time". Yesterday night, I briefly searched the web and found some guys
still attacking the "problem of time" but not in loop quantum gravity arena.
They're still in the context of geometrodynamics. (Gee, we should solute to
all those brave men who have attacked this problem!)
>
> It's worth keeping in mind that the Wheeler-DeWitt equation is firmly
> based on the notion that spacetime is a continuum. If spacetime is
> something discrete, we'll probably need some discrete analog of
> diffeomorphism-invariance, which will require some modification of
> the Wheeler-DeWitt equation.
>
Then it has to fix all our continuous groups/symmetries in QFT. Another
revolution ahead of the grand unification of quantum theory and gravity.
Maybe the Wheeler De-Witt equation implies a discrete structure behind the
continuum, but we don't have the concept/idea to formulate how an implicit
discrete structure at Planck scale dynamically attached to the spacetime
continuum. This is at least true for kinamatics, the eigenvalues of area and
volume operators built upon knot invariant states are discrete.
However, I sense a little bit of switching attitude, the attitude you showed
rebuting C. Francis about his "kludgy dicretization"of existing models. I
hope you don't really mean spacetime "is" truly discrete just because of that
equation. Otherwise, it is quite disastrous for our current theories because
all our differential concepts in GR, QFT... have to be replaced. Maybe black
hole singularity will be removed automatically in your discretized theory. In
addition, receovering continuous models from discrete models can be very
controversial or problematic like how you nailed down C. Francis inproper
intent or faults for his QED .
Since you mentioned discrete spacetime, hope you don't mind I put forward my
primitive thought. We shouldn't discard the continuum of spacetime because
it's too disastrous. If without sufficient evidence, we shouldn't shift our
paradigm. The area and volume operators built upon knot invariant states have
discrete eigenvalues and, on other hand, superstring theories have shown some
discreteness (The fate of spacetime by E. Witten in Physics Today) about
spacetime. So, it seems that there is forming a consensus that the continuum
of spacetime has an implicit fumdamental planck scale dynamical structure.
We should realize this consensus, i.e, finding evidence from different
aspects to substantiate this concensus. Now, like Rovelli and Reisenberger
have built spin foam on simplicial complexes. They have a step further this
thought. I wonder will they come up operators on spin foam have discrete
eigenvalues. On the other hand, in V. V. Batyrev a paper "Dual polyhedra and
Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties"
alg-geom/9310003, he has shown some combinatorial conditions for determining
the Calabi-Yau dual. Hence he conjectured the isomorphism of two conformal
field theories on the two polyhedra. We should search out these kind of
combinatorial/discrete structure (dynamical and kinamatical). We should
contrast these combinatorial structures with each other and find out any
similar analogous behind them. If we don't have any common basic math terms
to compare and contrast the two approaches, then create the necessary
definitions until the concepts are analogous or even compatible. For example,
Rovelli has formed a simple moduli space for spin networks with
intersections (even though not much structure there). How about these we
thicken the Feynman diagram in spin foam and search out what corresponding
meaning in its dual spin foam, etecetra like that in order to form a moduli
space. Then enrich the structure of this moduli to see whether we can bring
any different combinatorial concepts closer togehter.
I believe that the mother nature has a unique truth about its physical laws.
If nobody willing to form a consensus, then it turns out incompatible
mathematical structures or theories. Then in some sense, we're trying to
disprove each other. So, be conservative.
With regards,
K. M. Hui
John Baez
>In article <8cj8gj$buc$1...@pravda.ucr.edu>,
> ba...@galaxy.ucr.edu (John Baez) wrote:
>> In loop quantum gravity, spatial diffeomorphisms don't have any
>> infinitesimal generator [....]
>They don't? I thought there is something like a "loop space
>derivative", can't it be used for the purpose?
Oooh! There's an expert in the house! Someone who actually knows
about this stuff! Great!
Let me explain... and not just to you, but to the whole audience.
I'll start with a bunch of review material and conclude by actually
answering your question.
In loop quantum gravity we can take kinematical states to be certain
functions psi on the space of loops in the manifold S representing
space. We apply a diffeomorphism to such a state in the obvious way:
(g psi)(gamma) = psi(g^{-1} gamma)
where g is a diffeomorphism of S, gamms is a loop in S, and
g^{-1} gamma is the loop obtained by acting on gamma by the
inverse of the diffeomorphism g.
Now, suppose we have a vector v field on S. With a little luck,
this vector field will generate a 1-parameter family of diffeomorphisms
of S. Call it g(t). Now suppose we try to differentiate g(t) psi
with respect to t. If we can, we'll define
-i (d/dt) g(t) psi = C(v) psi
and we'll call C(v) the "infinitesimal generator" of our 1-parameter
family of diffeomorphisms. Then we say the kinematical state psi
satisfies the "diffeomorphism constraint" if
C(v) psi = 0
for all vector fields v (or at least all those which generate 1-parameter
families of diffeomorphisms). This constraint says that psi is invariant
under diffeomorphisms of space, and it's a close relative of the Wheeler-
deWitt equation.
So, what's the problem? Well, the problem is just that the derivative
might not exist! If it DOES exist, we can say
[(d/dt) g(t) psi] (gamma) = d/dt psi(g(-t) gamma))
and the quantity on the right is called a "loop derivative", because
it tells us how psi(gamma) changes as we start pushing the loop gamma
around using the diffeomorphisms g(-t).
In early work on loop quantum gravity, folks assumed this derivative
existed for the kinematical states of interest. But then Ashtekar and
Lewandowski constructed a very nice Hilbert space of kinematical states,
clearly "right" in many ways, but with the unfortunate property that the
derivative does NOT exist.
Now loop quantum gravity is split into two broad schools, which one
could call the "northern" and "southern" schools. The northern school
uses the Ashtekar-Lewandowski Hilbert space of kinematical states, and
gives up on using loop derivatives. The southern school attempts to
make sense of loop derivatives, and works with a space of kinematical
states with no clear Hilbert space structure. The main exponents of
the southern school are Rodolfo Gambini, Jorge Pullin and their
collaborators, mostly from South America. The northern school
includes Abhay Ashtekar, Carlo Rovelli and Lee Smolin.
It turns out that you can treat the diffeomorphism constraint even
when you can't take the derivative described above. You simply insist
that your states be invariant under diffeomorphisms! And you can make
progress this way....
It turns out that the nondifferentiability of kinematical states with
respect to diffeomorphisms is closely related to the "discreteness"
which shows up in the northern approach to loop quantum gravity.
So this issue is extremely important!
More precisely: if you insist on working with states psi for which
(d/dt) g(t) psi
exists, it becomes very hard to figure out the correct inner product
on kinematical states, and you can no longer say with such certainty
that areas and volumes take on a discrete spectrum of values. All the
rigorous calculations of the spectra of the area and volume operators
have been done in the northern approach.
Most of my work has been done on the nothern approach, but I think
it's too soon to say which approach is better. The southern approach
has a lot to do with the mathematics of "Vassiliev invariants" of
knots, and that raises some very interesting questions, like finding
a nice inner product on the space of Vassiliev invariants - but nobody
knows the answers to these questions. It could be that we are missing
some of the basic math needed to go further in the southern approach.
Maybe we'll figure this stuff out someday.