(It turned out that we exchanged messages in the wrong order in this
thread.
Hence you might want to first read my second
answer to Calvin's question.)
Viewing
message <aiok4u$chg$1@rs04.hrz.uni-essen.de>
From: Urs Schreiber (Urs.Schreiber@uni-essen.de)
Subject: time in ordinary QM (was Boosts)
Newsgroups: sci.physics.research
Date: 2002-08-06 18:01:35 PST
"Calvin Ritchie" <DonRitchie870@csWebmail.com> schrieb im Newsbeitrag
news:F9B19.11203$pg2.906940@bgtnsc05-news.ops.worldnet.att.net...
[...]
> There's a curious side issue related
to this involving commutation of
> time and Hamiltonian. Everyone knows that [H,t]=-i,
Hm, this is not true. Not in the naive sense, at least. There is a way,
however, to make it true in some sense and with some modification,
I think. See
below.
> but, when we write the
> 'time-evolution operator', T=exp(-iHt), and take a partial w.r.t.
t of an
> expression involving T (such as a Heisenberg rep operator), we always
let H
> commute with T. [...]
Comparing this situation with that of systems who have a Hamiltonian
*constraint* instead of a Hamiltonian generator of time evolution might
suggest
a way to resolve the issue.
Instead of treating the time parameter different from the other coordinates
of
configuration space one could instead "parameterize" the dynamics by
letting
"t" be one of the system's coordinates. In quantum mechanics it should
work
like this:
Define a Hilbert space of *kinematical states*, say that of rapidly
decreasing
complex values functions over space *and* time. If M is the configuration
space
of the original system and R is the time "axis" then this could be
S(R x M).
This space does not in general contain any "physical" states, which
I'll come
to in a moment, it does however support, among the usual operators
previously
defined on S(M), the canonical pair t, p_t = -i hbar d/dt which satisfies
[t,p_t] = i hbar .
The usual Hamiltonian H, when trivially extended from S(M) to S(RxM),
commutes
with t and p_t. It's special relation to these two operators is not
in the
kinematics but in the *dynamics*. Hence, on top of the kinematic description
that we have set up we now need to impose the dynamics. This is done
by
"projecting" (in a suitably generalized sense) out the physical component
of a
kinematical state. Since we want physical states to be those that satisfy
the
Schroedinger equation
(p_t + H)|phys> = 0
we define the Hamiltonian constraint
C = p_t + H.
In order for it to have any non-trivial solutions we must extend it
in the
usual way from an operator on S(RxM) to one on the dual space S*(RxM),
the
"extended kinematical state space" and physical states |phys> must
be elements
of this space.
So now we have a system, governed purely by the Hamiltonian constraint
C|phys> = 0,
which expresses the fact that everything must be independent of the
parameter
tau that we (here only implicitly) introduced to parameterize the system
and
make time an ordinary coordinate. This can be made precise by observing
that C
does indeed generate reparametrizations because exp(tau C) sends a
kinematical
state concentrated at one instant of coordinate time t_0 to one concentrated
at
t_0 + tau with the same projection onto all physical states.
Hence the constraint C|phys> = 0 expresses the fact that physical states
are
those supposed to be gauge invariant under reparametrization gauge
transformations. In light of this it is not surprising that one finds
that the
standard scalar product on physical states, by the above definition,
is
ill-defined, since naive evaluation yields infinite values. This is
because the
standard scalar product, as can be checked explicitly, integrates over
all
reparametrization-equivalent versions of a state. So one has to try
harder. The
solution is of course to define a suitable gauge-fixed scalar product
which
avoids the integration over gauge-equivalent states. In the simple
setting
discussed here this gauge-fixed scalar product is easily guessed to
be simply
integration over M at a fixed (but arbitrary) time t.
For me, this construction clarifies the relation between the Hamiltonian
and
the time parameter in QM. The Hamiltonian H is not really the canonical
conjugate to time, but p_t is. When restricted to the physical subspace
p_t|phys> = - H|phys>, but both are different objects.
But since all the above is really an exercise in "physics made difficult"
one
might just as well continue and use heavy machinery, for instance BRST
cohomology, to deal with the issue of gauge fixing. I have once tried
to do
this, because it helped me understand gauge fixing in supersymmetric
quantum
cosmology. By extending the state space once again, this time so that
it admits
a grading, one can construct a nilpotent BRST operator Q from our Hamiltonian
constraint C and study its cohomology. Since gauge fixed physical states
are
those annihilated by the constraint (annihilated by Q), but not themselves
the
result of a gauge transformation (not in the image of Q), they are
precisely
the elements of the cohomology of Q. This cohomology can be characterized
by
constructing a "coBRST" operator Q*, which is the adjoint of Q with
respect to
some, non-physical, positive definite scalar product. It is then known
by
general arguments that the cohomology of Q is the intersection of the
kernels
of both Q and Q*. So Q*|phys> can be regarded as the gauge fixing condition.
Using Q and Q* one can define the gauge fixed scalar product by standard
methods of BRST theory and it does indeed turn out to reduce to the
naive
ordinary scalar product. I give more details on this construction in
section
2.3.3 "BRST-cohomology of an ordinary Hamiltonian" in
http://www-stud.uni-essen.de/~sb0264/sqm.html. The point of this construction
is however its generalization to systems whose Hamiltonian is a Dirac
operator,
like supersymmetric theories.
--
Urs.Schreiber@uni-essen.de
Viewing
message <jxd49.1763$Ke2.154487@bgtnsc04-news.ops.worldnet.att.net>
From: Calvin Ritchie (DonRitchie870@csWebmail.com)
Subject: Re: time in ordinary QM (was Boosts)
Newsgroups: sci.physics.research
Date: 2002-08-08 11:45:17 PST
I had written:
>.... Everyone knows that [H,t]=-i,
to which Urs Schreiber replied:
>Hm, this is not true. Not in the naive sense, at least......
I was, I thought, just re-writing the "common
quantum prescription":
H(=P^0)=id/dt,
P^j=-id/dx^j, etc.
where d means partial, in the most naive sense. Do you also object
to the
second, P^j, expression? If so, I think that I understand part of your
objection, but let's pursue it a bit.
Most of your comments are over my head. Perhaps
you can pull me up a
bit, or bring the comments down bit, and answer and/or comment on the
following questions/points:
You say:
>Comparing this situation with that of systems who have a Hamiltonian
>*constraint* instead of a Hamiltonian generator of time evolution
might suggest
>a way to resolve the issue.
I don't understand what you mean by this distinction.
I understand that
P^0 is the generator of time displacements in the same sense that P^j,
j=1,3, is a generator of space displacements.
.....you then say
>Instead of treating the time parameter different from the other coordinates
>of configuration space one could instead "parameterize" the dynamics
by
>letting "t" be one of the system's coordinates.
In SR, don't we "always" treat time the same
(except for the
metric, -1,1,1,1) as the other coordinates?
It has always bothered me that, in quantizing
the fields, we use the
"equal-time commutation relations", which does seem to place time on
a
special footing. Is that what you're getting at in the remainder of
your
comments?
>Define a Hilbert space of *kinematical states*, say that of rapidly
decreasing
>complex values functions over space *and* time. ......
...plus a lot of things that I understand even less than the above.
In QFT, the one-particle fields are always
parameterized by x^mu,
mu=0,3, by way of the terms exp(iP_mu x^mu), aren't they? Doesn't this
mean
that we're working on a 4-D manifold (with the usual spacetime metric)?
Although I've read Flanders, Schutz, and Frankel
on differential
geometry, I've not yet gotten into topology beyond the mentions in
those.
Nor have I yet worked with the concepts in Frankel enough to feel
comfortable with them. Also, I've read Ryder (why did BB guns just
come to
mind?:-) and Weinberg on BRST, and the like, and can follow their
developments, although I haven't worked enough with them to see how
this
ties into the question of time in the present context.
I'll stop there for now and try to digest some more of your comments.
Submitted afternoon 7 August 2002,
Don Ritchie
DonRitchie870@csWebmail.com
Viewing
message <aj0ftr$duu$1@rs04.hrz.uni-essen.de>
From: Urs Schreiber (Urs.Schreiber@uni-essen.de)
Subject: Re: time in ordinary QM (was Boosts)
Newsgroups: sci.physics.research
Date: 2002-08-11 13:00:49 PST
"Calvin Ritchie" <DonRitchie870@csWebmail.com> schrieb im Newsbeitrag
news:jxd49.1763$Ke2.154487@bgtnsc04-news.ops.worldnet.att.net...
[...]
> I was, I thought, just re-writing the
"common quantum prescription":
> H(=P^0)=id/dt,
> P^j=-id/dx^j, etc.
> where d means partial, in the most naive sense.
Ah, I see. I misinterpreted the situation and so my comment couldn't
make sense to you. You are right, but I think, from looking back at
your original post, where you also used "H" as the generator of
parameter evolution, that you are mixing up two different cases which
need to be distinguished, namely that where time is a coordinate, a
degree of freedom of the system, and that where it is merely the
parameter of evolution. Both cases are related, but different.
That's
partly what I was trying to address in my last post. Let's try to sort
that out:
So what's mechanics all about?
(Heh, as you see I'll start from the very beginning. But I think it
will be worth it and I'll make it as brief as possible) So just
briefly recapitulating: We have some sort of physical system which
can
have some sort of configuration. We enumerate all theoretically
possible configurations by a set of parameters, the canonical
coordinates "x^i". These parametrize what is called configuration
space M. Associated to each canonical coordinate is a canonical
momentum "p_i". Canonical coordinates and canonical momenta together
are coordinates on the phase space over configuration space (which
is
technically the cotangent bundle over config space). There is the
Poisson bracket for functions on phase space, which, among other
things, says that
{x^i, p_j} = delta^i_j .
A single point in phase space is your system in some state and
prepared to move somewhere. For it to move there must be a parameter
along which to move (1). Let's call it tau. (Don't think of that
as
related to "time" yet, it's just a parameter.) The laws of classical
mechanics tell us that given some Hamiltonian, which is a *function
on
phase space* (assume it to be independent of tau, for simplicity),
we
can produce one-parameter families of points in phase space, the
"classical trajectories", by solving Hamilton's equations:
d x^i / d tau = d H / d p_i
d p_i / d tau = - d H / d x^i .
So these solutions are phase-space valued functions of tau and hence
necessarily one-dimensional objects.
That's how ordinary mechanics works. It requires precisely one
parameter of evolution, no matter what.
Now on to quantum mechanics. We realize that the above classical
evolution is really only a first approximation to reality and that
for
some largely unknown reason (but, ok, we can derive everything from
the path integral, for instance) we have to make phase space
non-commutative and all that. In particular the above ingredients are
transformed as follows (roughly, heuristically and with no more detail
than shall be needed below):
The space of states becomes a Hilbert space h, say h = S(M), the space
of complex valued functions square integrable and rapidly decreasing
over configuration space M. Every canonical coordinate x^i becomes
the multiplication operator by x^i on h and every canonical momentum
becomes the differential operator p_j = -i d_j .
Why am I saying all that, even though you know it already? Here is one
point I think is important to make: The partial derivative with
respect to our parameter of evolution, d/d tau, is *not* an operator
on the Hilbert space h. That's because the functions that are
elements of h are functions over configuration space M and hence
depend solely on the canonical coordinates x^i. There is absolutely
no tau-dependence that could be differentiated. The elements of h are
a quantum substitute (roughly) for the classical points in phase
space. You cannot differentiate a point in phase space by tau, either,
that would be nonsense. What you CAN differentiate with respect to
tau
in classical mechanics is the entire classical trajectory, the
one-parameter family of phase-space points, parameterized by
tau. Analogously, what you CAN differentiate with respect to tau in
quantum mechanics is a one-parameter family of elements of h,
parameterized by tau. Of course, such is obtained by solving the
quantum version of Hamilton's equations, the Schroedinger equation
d_tau psi(tau) = (-i) H(x^i, -i d_j) psi(tau) .
Note well here two things:
First: For fixed tau, say tau=12, psi(tau) = psi(12) is a fixed
element of the Hilbert space h. But psi itself is rather a function
psi: R -> h
tau |-> psi(tau)
that is not an element of h, but a one-parameter family of elements
of
h, a map from the reals into h.
Second: The Hamiltonian H in the above equation is a function on the
quantum phase space. It is represented (induced by the above
representation of the canonical coordinates and momenta) on h as some
differential operator. As such it knows nothing of tau. (I have
assumed it to be tau-independent in order to avoid the complication
that H itself is a one-parameter family of operators, which would only
make things look more difficult than they are and would not affect
anything of what I am trying to say here.) In particular, the
"commutator of H with tau" is not even well defined. The commutator
instead commutes two operators that act on h; but d_tau is not an
operator acting on h.
But you are perhaps thinking all along: What is this fuss about d_tau
not being a quantum operator. Don't we have an operator p_0 = -i d_t
in relativistic quantum mechanics?
Yes we do, and everything is fine. We just have to sort out what kind
of object -i d_t really is in "relativistic" quantum mechanics. The
answer is easy, even trivial.
I have taken pain to use the name "tau" for the evolution parameter
instead of the usual "t". Names of variables don't affect the
formalism, but they may affect our thinking about the formalism. The
usual presentation of the non-relativistic Schroedinger equation in
QM
textbooks as
i d_t psi = - (1/2m) d_x^2 psi + V psi
does not make it particularly clear that d_t enters the conceptual
framework here on a rather different footing than d_x does. In
particular, it is conceptually quite different from the d_t that
enters the Klein-Gordon equation as in
-(d_t)^2 psi + (d_x)^2 psi = m^2 psi .
How can that be? The reason is that in the context of the relativistic
Klein-Gordon particle the physical time "t" is not a parameter, but
a
canonical coordinate! Hence it has a canonically conjugate momentum
p_t, which is quantized as usual to yields p_t = -i d_t <=> p^t
= + i
d_t . (So is there no further parameter, as opposed to my above claim
that mechanics requires precisely one external parameter? I'll come
to
that in a moment.)
Let's see in more detail why this is true:
Hamiltonian formalism is fine, but now we need a Lagrangian first: The
Lagrangian of the massive relativistic point propagating in Minkowski
space with metric eta_mn is
dL = - m ds .
Here m is the mass of the point and s is the proper distance along the
worldline of the point as it moves through Minkowski space (its
"world-volume").
While that's a nifty definition, as it stands it does not allow us to
calculate anything. To do that we have to measure the proper distance
somehow and to do that we need a coordinate on the point's
worldline. With hindsight, this coordinate shall be called tau. (Now
guess what...) So every point on the worldline is continuously and
monotonically indexed by a value of tau and we can calculate the
proper distance by the following well known expression
(ds)^2 = eta_mn dx^m dx^n
=>
ds = sqrt( -eta_mn d x^m / d tau d x^n / d tau ) d tau .
(Here the minus sign is because my eta is "space-like", i.e
eta = diag(-1,1,1,1).)
Comparing this Lagrangian with that of a non-relativistic point of
mass m in Euclidean space, which reads
dL = ( delta_ij d x^i / dt d x^j /d t ) dt
one sees, as advertised, that "tau" and "t" play analogous roles. One
also sees that the Minkowski coordinates x^n(tau) (n=0,1,2,3) are the
canonical coordinates of the relativistic point. In fact, the
configuration space of the point is identical to physical space
itself. (But that's a coincidence. It does not hold for systems
consisting of 2 or more points or for fields.) In particular, x^0 =
t,
the time coordinate of the point, is a *canonical coordinate* of this
system. It is not the evolution parameter. Tau is. Tau is the
arbitrary parameter that parameterizes the classical trajectories in
configuration space, which in this simple case is just the same as
the
classical trajectory in physical space. Note also that, while tau
monotonically increases along the world-line, the point may a-priori
move in any direction in configuration space, in particular it may
a
priori take non-monotonically increasing values of x^0. We have to
look at the equations of motion to see if it really does so, though,
but it is an important principal point that arises when physical time
is taken as a canonical coordinate (a degree of freedom of the system)
instead of as the evolution parameter.
To every canonical coordinate there is a canonical momentum p_n(tau)
with the Poisson bracket
{x^n, p_m} = delta^n_m .
Note how I put the indices upstairs on x^n and downstairs on
p_n. That's the way to do it and it's due to the momenta being really
elements of the cotangent space. When you shift indices by means of
the Minkowski metric you get
{x^n, p^m} = eta^nm .
For possible later generalizations to curved space it is crucial to
keep in mind that the p_n come with covariant (down-stairs) indices
and that it is a non-trivial operation to raise them by means of the
metric.
But we want to canonically quantize and so we want a Hamiltonian. To
get that we need the canonical momenta in terms of the
"tau-velocities"
v^n = d x^n / d tau
by way of
p_n = d L / d v^n .
One finds
p_n = m v_n / sqrt(-v^n v_n) .
So we happily go along and compute the Hamiltonian as
H = v^n p_n - L
= m v^n v_n / sqrt(- v^n v_n) + m sqrt(- v^n v_n)
= 0
Oops! Now that's something, the Hamiltonian identically vanishes!
I don't
know if this surprises you, perhaps you know all that, but otherwise
it may
reassure you to know that the equation
H = 0
in a somewhat more involved context, namely that where the original
action is not that of a point in spacetime but that of the quite
analogous situation of the entire universe in *its* configuration
space, has made a lot of people write considerable amounts of paper
full of erudite stuff. However, there is actually no mystery
here. This, interestingly, becomes quite clear in the quantum theory
-once we have figured out to construct this from the strange
Hamiltonian above.
The clue to progress is to have a closer look at the canonical
momenta. They were found to be
p_n = m v_n / sqrt(-v^n v_n) .
Usually the momenta can, a priori, take all kinds of values. But these
cannot. To see this take their square
p^n p_n eta^nm p_n p_m = m^2 eta_nm v^n v^n / (-v^n v_n) = -m^2 .
Hence, due to the normalization factor 1/sqrt(-v^n v_n) they carry,
the canonical momenta sort of always dynamically adjust so as to
square always to -m^2. Because of
p^n p_n = -(p_0)^2 + (p_i)^2 = - m^2
this defines a hyperbola and one says that the momenta have to be "on
mass shell", or simply "on shell". But this means that the system
we
are studying here is not free to explore its phase space but that it
is *constrained* to be on-shell and to satisfy p_n p^n = -m^2 . This
wasn't manifest in the original Lagrangian and because we ignored it
we ran into the strange situation of finding an identically vanishing
Hamiltonian.
The remedy, to make a long story short (hmm...), is to replace our
original Lagrangian with one that is classically equivalent, in that
it produces the same equations of motion, but also explicitly
incorporates the constraint. The way to do that is by introducing a
new independent canonical coordinates, N, which plays the role of a
Lagrange multiplier.
Therefore consider now the Lagrangian
L = p_n v^n - N(p_n p^n + m^2)/2 .
Varying it with respect to N gives the mass-shell constraint, by
construction, and varying with respect to x^n gives
d p / d tau = 0,
which is again the classical equation of motion. So this is indeed
equivalent to our original Lagrangian. (This is also evident when you
insert the definition of p_n into the above expression for L.) Note
that now
H = N(p_n p^n + m^2) / 2,
is not *identically* zero, even though the equations of motion demand
that it be zero.
With the Hamiltonian at hand the system may finally be canonically
quantized, but since I have already written so much I can just as well
take the time to quickly point out what all this constraint business
has to do with the original issue of tau being a parameter along the
worldline:
Namely from the above Hamiltonian one finds
d x^n / d tau = N p^n
and from this
dL = (1/2)( (1/N) (d x^n / d tau) (d x_n / d tau) - N m^2 ) d tau .
Note that N is so far a free variable, but in the combination N d tau
it looks a lot like a measure on the worldline of the
particle. Indeed, let's define a metric on the worldline, with only
one element
g_(tau tau)
and identify the volume element
dvol = sqrt(-g_(tau tau)) d tau
with N dtau, so that
N = sqrt(- g_(tau tau)) .
This way the above Lagrangian takes the form
dL = (1/2) (g^(tau tau) d_tau x^n d_tau x^m eta_nm - m^2) dvol .
This is much handier than the original square-root Lagrangian but also
more illuminating. It makes it manifest that the freedom we have in
choosing our parameterization tau of the worldline of the particle
is
that of choosing a metric on the worldline.
Ok, so much for the digression, now on to the point: We had found the
Hamiltonian
H = N/2 (p_n p^n + m^2) .
The p_n are canonical momenta, so we know how to quantize them
p_n = -i d_n = -i d / d x^n .
This gives
H = H(x^n,p_m) = N/2 (- eta^mn d_n d_m + m^2) .
Following the general prescriptions for doing canonical quantum
mechanics that I outlined at the very beginning, we have the
Schroedinger equation
i d/d tau psi = N/2 (- eta^mn d_n d_m + m^2) psi
that describes how wave functions psi, (elements of a Hilbert space
h
consisting of functions over configuration space which is space*time*
in our case) evolve along with the evolution parameter tau. But we
know that tau has no physical meaning. It is an arbitrary metric on
the worldline of the particle. If we change the parameterization of
the worldline we do not want the physics to be affected by that at
all! So is the above Schroedinger equation meaningless? No, it must
be
correct if quantum mechanics is correct. The point is, however, that
not *all* its solutions are acceptable, because our system is subject
to a constraint, the mass-shell constraint. This says classically that
p^2 + m^2 = 0, which quantum mechanically means that
(- eta^mn d_n d_m + m^2)psi = 0 .
This in turn says that the Hamiltonian has to annihilate physical states
H(x^n, -i d_n) psi = 0 .
Therefore the Schroedinger equation in terms of tau is quite alright,
it needs however be accompanied by the restriction that only the
subset of its solutions which also satisfy the above constraint are
physically realized. This gives
d/d tau psi = 0
(- eta^mn d_n d_m + m^2)psi = 0 ,
and that's good, because the first equation (the left hand side of the
Schroedinger equation) now says that every physical state must be
independent of the arbitrary metric that we put on the worldline,
while the second equation is just the Klein-Gordon equation!
Pooh! I didn't expect it would take me so long to make this point. :-)
But anyway, I hope this makes it quite clear that the p_t = -i d_t
operator that appears in the Klein-Gordon equation is a *canonical
momentum* operator which acts on a Hilbert space of functions over
configuration space, where in this case the configuration space does
include the physical time coordinate as a degree of freedom. It is
not, however, the derivative with respect to the evolution parameter
tau.
Note also that
[x^0, p_0] = [t,-i d_t] = i
[x^0,H] = iN
[x^i, H] = -iN
and, if you wish
[tau, H] = 0
but the last equation does not really make sense, as I discussed above.
The situation is superficially different when you look at
non-relativistic quantum mechanics in the usual formulation, because
then what is written as d/d t is what I have written as d/ d tau and
one must not confuse the d_t derivative in non-relativistic QM with
the d_t derivative as in the above treatment of the Klein-Gordon
equation.
The whole point of my previous post was actually to demonstrate (not
that this is news to the experts) that the kind of "parameterized
dynamics" that is encountered in the classical and quantum mechanics
of the *relativistic* point particle is *not unique to relativity*,
but that any boring-old non-relativistic system may be parameterized
by introducing a coordinate on the system's world-line and making
physical time one of the canonical coordinates.
> In SR, don't we "always" treat time
the same (except for the
> metric, -1,1,1,1) as the other coordinates?
> It has always bothered me that, in
quantizing the fields, we use the
> "equal-time commutation relations", which does seem to place time
on a
> special footing. Is that what you're getting at in the remainder
of your
> comments?
I really wanted to address this issue, but now I am too tired. Maybe
another time. But you can also figure out the answer yourself: The
essential idea is that an ordinary field-theory Lagrangian on a
*fixed* spacetime, say on Minkowski space, gives you a
*non-relativistic*-type of Schroedinger equation for the fields, with
no constraints and no arbitrary parameterization. Field theory is
really no different from quantum mechanics in this respect. You can
consider the quantum mechanics of a single non-relativistic particle
as a field theory on the particle's wordline (with fixed
parameterization), which has "coordinate field" propagating on
it. This is one way to see why you get "equal-time commutation
relations" in field theory, just like you also have equal-time
relations in quantum mechanics. To do away with this you again have
to
parameterize the field's dynamics and ensure that everything is
independent of the parameter. Since a field is to spacetime as a
particle's coordinates are to its wordline, this means that you have
to introduce a *variable metric on spacetime*, just as I did above
for
the particle-worldline. If you then vary the metric on spacetime you
get a Hamiltonian constraint which is then a Klein-Gordon type of
equation. If you do that and also allow for a term sqrt(g)R in the
action of the fields on spacetime, to make the metric really
dynamical, you get general relativity coupled to your fields. One can
do essentially the same construction as for the free relativistic
point particle for this huge system, called the ADM approach. One can
write down the quantum Hamiltonian of it, which will look like a
Klein-Gordon equation. It is not well defined due to
non-renormalizability and all other kinds of problems, but you can
certainly write it down just to see how everything follows the same
pattern.
[...]
> I'll stop there for now and try to digest some more of your comments.
Forget the comments from my first post until you have read the
comments in THIS post! :-)
(1) There are formulations of mechanics which do not involve precisely
one parameter of evolution. For instance you can find recent papers
by
C. Rovelli on the arXive where he is developing field theory with more
than one parameter. But that's not the canonical way to do it.
Viewing
message <ajhbob$f0i$1@rs04.hrz.uni-essen.de>
From: Urs Schreiber (Urs.Schreiber@uni-essen.de)
Subject: Re: time in ordinary QM (was Boosts)
Newsgroups: sci.physics.research
Date: 2002-08-19 19:49:31 PST
We had been talking about the free relativistic point particle, which
is an
example of a parameterized system, and how its quantization gives rise
to the
Klein-Gordon equation. We had mentioned some aspects of field theory
but have
not yet gone into that. In particular, Calvin Ritchie was concerned
about the
appearance of "equal-time commutation" relations in relativistic physics:
"Calvin Ritchie" <DonRitchie870@csWebmail.com> schrieb im Newsbeitrag
news:gYz59.7688$Ke2.648503@bgtnsc04-news.ops.worldnet.att.net...
[...]
> I think I see where you're going with
the "equal-time commutation", and
> will try to work it out for myself. If I have trouble, I'll yell.
If you
> want to expand on it in the meantime, I'll guarantee you at least
one very
> interested reader.
I have the weird plan to explain this using brane physics.
The answer to "Why equal time commutation relations in relativistic
field
theory?" is roughly "Because we have deparameterized the field theory
by
identifying physical time with the parameter of evolution. When we
reintroduce
an independent evolution parameter, then we instead get 'equal parameter
commutation relations', in much the same way as we have 'equal parameter
commutation relations' for the relativistic point particle." Ultimately
this
amounts to discussing the Hamiltonian (ADM) formulation of general
relativity
coupled to some fields. I don't want to do that right now. But I do
want to
make some progress in that direction by building on our expertise with
the
Hamiltonian formulation of the free relativistic point - which is just
the free
0-brane.
So what I want to do is take the action functionals from my last post
in this
thread and straightforwardly generalize them by enlarging the dimension
d of
the world-volume of the object they describe. (Recall that for the
point
particle this world-volume, namely its world-*line* was (d=1)-dimensional.)
The
resulting action is that of a free relativistic p-brane, where p=d-1.
I'll make
an ADM split of the metric on the brane and obtain the ADM-Hamiltonian.
The point of all this is that
1) It's fun.
2) It's easy.
3) It gives further insight into the relation between physical
time and
unphysical parameters.
4) It is a direct preparation for the Hamiltonian formulation
of general
relativity.
I should admit that I am only beginning to learn about branes myself
right now.
So I cannot completely rule out that I am subject to some beginner's
misconception. While I am rather convinced that I have the (easy) calculations
right, it may well be that I fail to properly use some terminology
or misjudge
the true meaning of some formulas. With some luck there'll be an expert
willing
to chime in and fix such things.
After this lengthy introduction I am finally ready to begin:
First we need to adjust our brains to branes. I mean, we need to get
straight
what we are trying to describe and how that is to be forged into formulas.
What were the ingredients for the description of the relativistic point?
First,
there was the point's world-line, a (d=1)-dimensional manifold (a line)
which
was parameterized by a real number tau. We saw that one is more or
less forced
to consider this manifold to be equipped with a metric. In one dimension
a
metric is a pretty trivial thing, it is a tensor with only a single
entry
which, I believe, I had called g_(tau tau). This is too clumsy a notation.
We
will soon think of tau as the 0th coordinate on the world-volume, so
that I now
write
g_(tau tau) = g_00 .
This metric was related to a parameter N by
g_00 = - N^2 .
So this was one ingredient: the wordline equipped with a metric. This
shall be
called the *base space* B from now on. The name is due to the fact
that this
space is the base for the second ingredient, namely the coordinate
fields.
The worldline is not just sitting there, but we imagine it to be embedded
into
physical spacetime. So every point of the wordline has D spacetime
coordinates
X^mu associated with it, where D is the number of spacetime dimensions.
Hence
these X^mu are functions from the worldline B=R (the reals) into physical
spacetime T
X^mu : B -> T .
tau |-> X^mu(tau)
The "fields" X^mu are called the *embedding fields* because they embed
the
worldline into spacetime. Maybe this looks like a particularly convoluted
way
of thinking about an ordinary trajectory in spacetime, but it turns
out to be
very useful.
Finally, the third ingredient, which has already made its appearance,
is
spacetime T itself. However, erudition forces us to invent a name that
better
captures the true nature of the concept involved here, so T shall be
called the
*target space* from now on, since it is the space in which the embedding
fields
take values. The important thing is that the target space comes equipped
with a
metric tensor, too. Since I am using capital letters for target space
objects,
this metric shall be called G_mu nu . It is interesting to consider
metrics
G_mu nu of any sort, but before doing that we should have mastered
the easiest
case, which is G_mu nu = eta_mu nu, the flat Minkowski metric. Hence
this is
what I assume from now on. Also, we will be mostly concerned with properties
of
the base space and then all these target space indices will be a nuisance
since
we will mainly have to deal with base space indices. Therefore I decree
the
convenient shorthand
X^mu Y^nu G_mu nu = X .Y
for any two objects X, Y which carry target space indices.
One might think, looking around oneself, that the dimension of target
space, D,
should be equal to 4. But actually most people that you'll encounter
who know
about branes will insist that D=10 or D=11. For our present purposes,
however,
the exact value of D is completely irrelevant and I'll leave it unspecified.
In
fact, the whole machinery that I am describing here can also be applied
to the
case where the base space coincides with physical spacetime and where
the
target space is some abstract space which may well have any dimension.
This is
in fact the case for ordinary field theory, which, however, I do not
consider
here.
Looking back over the last paragraphs, what has been accomplished? We
have
realized that the theory of the relativistic point that we had been
considering
is conceptually the theory of a set of embedding fields that map some
base
space into some target space:
|
. . .. .. ... /... . . .
| X^mu . . .
. . . . . .|. . . . .
| ---------> target space
/
|
. . .. .. . .. ./. . ... . .
|
. . . . . . . . |. . . . .
base
space
Here the irregular line on the right is supposed to indicate the image
of the
base space under the embedding induced by the embedding fields X^mu.
For the
point particle this is just the trajectory in spacetime. This way we
are
distinguishing between the world-line of the particle in its abstract,
platonic
form, being just a line with a metric g_mn (the base space on the left)
and its
physical realization as a trajectory in spacetime, i.e. in target space
with
metric G_mu nu.
It is now extremely easy to generalize this setup to higher dimensional
base
spaces, which is the motivation for why I am going through all this.
You can
guess what one has to do, but I'll briefly spell it out:
Let base space B be a d dimensional manifold for any d>0. Assume that
this
manifold has the topological structure
B = R x Sigma ,
where R indicates the real axis and Sigma is some compact (p=d-1)-dimensional
manifold. We think of Sigma as our p-brane and of R x Sigma as the
world-volume
of this p-brane as it evolves along a parameter varying on the R factor.
So in
the case of the point particle one has Sigma = {.}, the set containing
a single
0-dimensional point, and B = R x {.} = R is the point's worldline.
If we set
Sigma = S^1, the circle, then B = R x S^1 is a cylinder. This is the
world-*sheet* of a closed 1-brane, a closed string. If we set, for
instance,
Sigma = T^2, the torus, then B = R x T^2 is the world-volume of a closed
2-brane, or membrane, of genus 1. And so on.
For simplicity I'll assume that base space B can completely be
parameterized
by a single coordinate patch with coordinates x^m, m = 0,1,..p . The
metric
tensor with respect to this coordinate patch is g_mn. (For the point
particle
we had p=0, x^0 = tau, and g_00 = -N^2 .)
For a general p-brane the coordinate fields are functions of the d base
space
coordinates x^m
X^mu = X^mu(x^m)
and their values determine how the p-brane moves through target space,
i.e.
through spacetime.
One might wonder whether the two metrics, g_mn and G_mu nu need to be
related
in any way. The answer is yes, which is a result of the physical dynamics
to
which I now, finally, come:
So far we have set up the (classical) kinematics of a p-brane, i.e.
we have
specified which degrees of freedom there are. Now we need to impose
the
dynamics by defining an action S. The action needs to be a function
that gives
a real number for every field configuration on the brane. The fields
are the
coordinate embedding fields and hence the action functional needs to
assign a
real number to every embedding of the brane history B into target
space/spacetime T. If you think about it, there are not so many natural
choices
for such a function. For the point particle we used
S = -m Vol_0 ,
where Vol_0 means the proper volume of the point's worldline, i.e. its
proper
length, as measured with the spacetime metric.
Vol_0 = integral over worldline of sqrt(- d_0 X . d_0 X ) dx^0.
Note that in this post I am using "d_n" for the partial derivative with
respect
to the n-th coordinate x^n on the base space. With my above convention
concerning target space index contractions one has
d_0 X . d_0 X = d_tau X^mu d_tau X^mu G_mu nu = ds^2 / d tau^2 .
The natural idea is to straightforwardly generalize this action principle
to
arbitrary p-branes, which have world-volume Vol_p, by setting the general
action proportional to the brane's world-volume:
S = - T Vol_p .
One conventionally replaces the constant factor m by the constant factor
T,
which is then called the brane "tension". For p=0 one has T=m. T is
just some
dimensionful constant that sets the scale of everything. It indicates
that we
are doing physics and not simply mathematics, but it is not essential
for the
following manipulations.
How does one measure the world-volume for p>0? The rule is: Take a tiny
speck
of the brane's history in spacetime (say of the worldsheet cylinder
for p=1)
with edge lengths dx^m for m=0..p . If the speck were flat, then its
volume
would simply be the product dx^0 dx^1 ... dx^p. Since it is not flat
in general
we have to account for the deformation of this volume element which
one can
show amounts to multiplying by a factor of sqrt(-h), where h is the
determinant
of the *induced metric* h_mn on the brane's spacetime history. What
is the
induced metric? It is just the metric which gives the length of a tiny
line
element of the brane history as measured *with the target space metric*:
h_nm = (d_n X) . (d_m X)
This way the target space induces a metric on the embedded manifold
(technically this is called the "pullback" of G_mu nu to B).
Hence an infinitesimal element of the brane's worldvolume has the proper size
dvol = sqrt(-h) dx^0 dx^1 ... dx^p .
and therefore the above action of the free p-brane reads
S = -T integral over B of sqrt(-h)
Conceptually there is nothing more natural than this action S. Yet,
as for the
point particle, it may be a little awkward to actually work with. One
might
rather have an action functional that involves quadratic expressions
in the
field's derivatives.
I'll call the above action the Nambu-Goto action. As far as I can see
this term
is usually reserved for the case where p=1, but here I want to use
it for all
p>0.
If you recall how we modified the point-particle's action from the Nambu-Goto
form to the form which involved the Lagrange multiplier N, then it
is
relatively easy to see what has to be done for general p to achieve
something
analogous:
Note that the Lagrange multiplier, N, turned out to be related to the
metric
g_00 on the world-line. So the idea might be to use the metric g_mn
in general
as some sort of Lagrange multiplier. Let's try this:
S' = -T/2 integral over B of sqrt(-g) ( g^mn h_mn + Lambda ) .
Note that now there are, as before, the embedding fields (hidden in
h_mn),
which are the physical degrees of freedom. But in addition there are
now also
the components of the base space metric entering the action, roughly
analogous
to Lagrange multipliers, and I have introduced a *constant* term
Lambda. Since
this term is multiplied by sqrt(-g) it is, technically, a so-called
*cosmological constant* on the brane. If you haven't encountered this
before
just accept the terminology, it is not supposed to mean anything deep.
But why do I introduce this term? Because it is needed to make the two
action
functional S and S' classically equivalent, and that's what I am aiming
to
achieve:
To see this we need to look at the equations of motion for g_mn that
derive
from S'. To find them one needs to know the variation of the determinant
of the
metric with respect to g^nm . This turns out to be
delta sqrt(-h) = -(1/2) sqrt(-g) g_nm delta g^nm .
Using this relation we find
0 = delta S' / delta g^mn
= sqrt(-g)( h_mn - (1/2)g_mn( g^mn h_mn + Lambda )
) .
Staring at this equation for a while one realizes that we need
Lambda = 2 - d = 1 - p
and that then the above equation of motion implies
g_mn = h_mn,
i.e. that the metric on the base space is just the metric induced by
the target
space metric.
I'll just show that this does work: For g_mn = h_mn we have
g^mn h_mn = d
and the above equation of motion is actually solved with Lambda as given
above.
But when g_mn = h_mn then the primed action becomes
S' = -T/2 integral of sqrt(-g)(g^mn h_mn
- Lambda)
= -T integral sqrt(-g)
= -T integral sqrt(-h)
= S,
and then both action functionals are classically equivalent. (Look at
this for
p=0 to see that it reproduces the analogous derivation for the point
particle.)
Hence the action that I will be concerned with in the following is
S' = -T/2 integral over B of sqrt(-g)( g^mn h_mn - d + 2 ),
and I'll call this the Polyakov action here, even though, again, this
term
seems to be reserved for the case p=1.
When you do this on a sheet of paper you should want to replace h_mn
by its
definition in terms of the fields
h_mn = d_m X . d_n X
to see that this action has a very familiar look. It just describes
a set of
ordinary scalar fields propagating on a curved manifold with metric
g_mn. And
this is why I can claim that this brane-business can bring us closer
to
understanding general relativity coupled to some fields.
There is possibly more truth to that I can make plausible in the present
context. A currently active research field is so-called "brane-cosmology",
which assumes that the 3-dimensional space that we seem to inhabit
*is* a
3-brane which propagates in some higher dimensional spacetime.
Ok. We are essentially doing field theory on the brane now. Our next
task is to
find the Hamiltonian associated with the Polyakov action for general
p. The
Lagrangian density that we are dealing with is
L = -T/2 sqrt(-g)(g^mn h_mn -d + 2) .
We just have to apply the usual prescription which says that one first
must
calculate the canonical momenta
P = delta L / delta d_0 X
= -T sqrt(-g) g^0n d_n X
(I'll leave the target space index of X^mu and P_mu implicit) and then
compute
the Hamiltonian density as
H = P d_0 X - L
and express all occurrences of d_0 X by the expressions in P. When you
write
that down you find a sum of several terms where various components
of g_mn
appear all over the place. It is not very illuminating. But one can
exhibit the
elegance of this Hamiltonian by writing the metric tensor g_mn in a
particular
way. This is called the ADM split:
The idea is simple: Since base space factors as B = R x Sigma it is
natural to
split the line element on B into parts that come from translations
on Sigma and
parts that come from translations on R. One therefore writes
ds^2 = -N^2 (dx^0)^2 + (N^i dt + dx^i )(N^j dt + dx^j) g_ij .
Looking at this expression you see that N, the same N as we had before
for the
point particle, which is called the "lapse function", measures how
much "proper
time" elapses as you move forward in "parameter time" (along the R
factor),
while the N^i, which constitute what is called the "shift vector",
measure how
much the "spatial coordinates" shift below your feet as you move along
R.
Finally g_ij is the metric purely on the spatial factor Sigma. There
is a
subtlety here when talking about space and time. Actually, only when
we are
on-shell, i.e when the equations of motions hold so that g_mn = h_mn
does this
make proper sense in base and target space.
When the metric in the above form is inserted into the Hamiltonian the
latter
simplifies somewhat. In particular it turns out that N and N^i can
be factored
out from all terms, so that they take their places as Lagrange multipliers,
as
expected. In ordinary field theory this would give the Hamiltonian
in its final
form. However, in the present case the Hamiltonian can be made to look
still a
little more elegant by replacing
N -> N / sqrt(k) .
where I have defined k to be the determinant of the spatial metric
k = det(g_ij).
After doing this there are no square roots of k left in the Hamiltonian.
If we
again use the equations of motion to set
d_i X . d_j X = g_ij
then finally the Hamiltonian takes the pretty form
H = N (1/2)( P^2 / T + T k ) + N^i (P . d_i X) .
This is the central equation of this post. Recall the various abbreviations
that enter it
P^2 = P.P = P^mu P^nu eta_mu nu
P . d_i X = P^mu d_i X^nu eta_mu nu
k = det(g_ij)
To better understand this Hamiltonian I go through the first few values of p:
p=0: This is the relativistic point that we already know. There is no
shift
vector on a point, N^i = 0 and no metric, k = 1. Also, as mentioned
above, the
"0-brane tension" is just the point particle's mass. Thus we have
H = N/2 (P^2/m + m),
as expected. Quantizing this gives the Klein-Gordon equation, as has
been
discussed.
p=1: This is the relativistic string. Now new effects appear. The first
thing
to note is that now we are really talking field theory on the brane.
All fields
in the Hamiltonian depend on the evolution parameter x^0 = tau that
is also
present for the point particle, but furthermore they depend on the
spatial
parameter x^1along Sigma = S^1, i.e. along the string. Hence by varying
N(x^0,x^1) at every point x^1 we get an infinite number of Hamiltonian
constraints
( P(x^1)^2 / T + T k(x^1) ) = P^2 /T + T (X')^2,
(where X' = d_1 X is the spatial derivative of the fields), one at every
point
of the string. This can, roughly, be interpreted as the dynamics of
an infinite
collection of relativistic points as in the p=0 case, however now with
a mutual
interaction which holds all these points together to form the string.
If we
forget that spacetime here is target space and instead look at base
space as
our "spacetime", then this is the first case that allows to address
the issue
of "equal time commutation relations" in quantum field theory. Here,
obviously,
we have "equal tau commutation relations". But tau, the temporal coordinate
on
the string's worldsheet, is just an arbitrary parameter. As for the
relativistic point, no quantum state may depend on tau. Hence it is
completely
unessential that commutations relations are fixed to some tau value.
More
precisely, take any operator A and evolve it along tau with the quantum
version
of H_0 = P^2 /T + T (X')^2
A(tau) = exp(i tau H_0) A(0) exp(-i tau H_0) .
Since all physical states are annihilated by H_0 the expectation value
of
A(tau) is the same for every tau
<A(tau)> = <phys|A(tau)|phys>
= <phys|exp(i tau H_0) A(0) exp(-i tau H_0)|phys> = <phys|A(0)|phys>
.
The fact that I used only H_0 in this example shows that another thing
is new
now that the spatial hyperslice Sigma has a non-vanishing extension:
There is a
further constraint. Namely by varying N^1 in the full Hamiltonian we
find that
P . d_1 X
also has to vanish at every point of the string, or, if you wish, of
our
cylinder spacetime. It can be shown that this expression generates
reparameterization not along tau = x^0 but along x^1. These transformation
are
called diffeomorphisms. Looking back at the full Hamiltonian makes
it clear
that there is one diffeomorphism constraint (at every point and) for
every
spatial parameter x^i. This pattern remains true for every reparameterization
invariant field theory, in particular for general relativity: In Hamiltonian
formulation the dynamics is completely determined by the temporal
reparameterization and the spatial diffeomorphism constraints.
p = 2: This is the relativistic membrane. Conceptually, there are no
new
phenomena here. We again have the temporal reparameterization and the
spatial
diffeomorphism constraints, now two of them. As soon as we know to
handle one
spatial dimension we can handle more of them, in principle, so that
poses no
new problems. But the membrane is far from boring, and that's still
an
euphemism.
When dealing with these extended objects one is soon and often reminded
of the
fact that geometry in 2 dimensions is somehow very special. That's
why the
string is special, its wordsheet is 2-dimensional. And the membrane
is special
because its spatial extension is 2-dimensional. This gives rise to
the
following intriguing facts:
In two dimensions there is a particular way to rewrite the determinant
of the
induced spatial metric as the product of two "symplectic brackets"
{. , .}
k = {X^mu,X^nu}{X_mu, X_nu}/2
where
{X^mu, X^nu} = epsilon^ij d_i X^mu d_j X^nu .
These are called Nambu brackets. With this identical reformulation the
membrane's Hamiltonian now reads
P^2 / T + {X^mu,X^nu}{X_mu, X_nu}/2 .
One can invent similar brackets as above for any dimension and rewrite
the
general p-brane Hamiltonian in this form. But p=2 is special because
only here
can {X^mu,X^nu}, which is an operation between continuous coordinate
fields, be
approximated, in some suitable sense, by the commutator of ordinary
*matrices*.
It can be shown that by replacing the coordinate embedding fields X^mu
in the
entire theory by large square n x n matrices one obtains some sort
of
regularization of the original theory, i.e. a theory that approximates
the
originally infinite number of degrees of freedom (D coordinate fields
at every
point on Sigma) with a large but finite number of degrees of freedom
(the
components of D matrices). This reduces all membrane theory to quantum
mechanics and this is essentially what is known as "Matrix Theory".
Now,
"Matrix Theory" sounds innocent enough, but there are serious suggestions
that
something very deep is hidden in this theory, somehow, namely nothing
less than
M-theory itself, the hypothetical non-pertubative version of string
theory.
I find it fascinating how the simple idea of generalizing the natural
point-particle action S = -m Vol to higher dimensional objects by the
most
natural and compelling generalization of Vol to arbitrary proper volumes
gives
rise to a hierarchy of theories of p-branes that somehow seems to be
the
backbone of all of string theory. You start at p=0, you take the very
first
exit and find ordinary relativistic quantum mechanics. You take the
next exit
and find that to use the quantum lane you are not free to move much
but have to
follow the route prescribed by a number of consistency conditions which
lead
you right to a unique universe of new concepts. If you instead take
the exit at
p=2 you find something that first looks totally different, but after
cruising
for a while you find yourself in precisely the same region that the
exit at p=1
led to. And all that simply by thinking S = -m Vol to the end.
P.S.
I'll be on vacation for the next three weeks, with probably only very
limited
internet access. So don't expect any responses from me during that
time.
--
Urs.Schreiber@uni-essen.de
