I have been studying tensor calculus in relativity a little bit before
school starts, and I have a question that I haven't been able to answer
by myself. I know that we can rewrite Maxwell's equations using the
electromagnetic field tensor,
F^(mu nu) =
(0 E^1 E^2 E^3)
(-E^1 0 B^3 -B^2)
(-E^2 -B^3 0 B^1)
(-E^3 B^2 -B^1 0)
And we can re-write Maxwell's equations as, with d meaning curly-d, and
J is the 4-current
d_nu F^(mu nu) = 4 pi J^mu
d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) = 0

With which we can do all kinds of nice special relativity-stuff, and, I
presume, general relativity stuff, too (although I haven't gotten that
far in what I've been studying).

Now, I know that in quantum mechanics, we have a complex valued state
vector Psi, satisfying some differential equation (which we'll assume
is not the Schrödinger equation, but is relativistically correct,
because I've been thinking about relativity). We can write Psi in
terms of real and imaginary parts, say Psi = A + i B. Now, presumably
if we wanted we could re-write the differential equation that Psi
satisfies as two separate differential equations, one in terms of A and
one in terms of B. Now, it seems like we could form another "quantum
mechanical field tensor" in terms of the components of A and B, just
like we did for E and B, and re-write our differential equations as
tensor equations like we did before, and do more relativity-stuff.
Now, it seems to me that since we've got something in the tensor
language of general relativity, we should be able to do general
relativity with this. But, obviously, we can't, or people would be
doing this. Why don't we do this (or do we, and no one has told me
about it)? At what point does this break and not make any sense
anymore? Can we at least do relativistic quantum mechanics like this
if we wanted to?

I'd really appreciate any comments anyone has about this!

Thanks,
Jeremy Price

cfgauss wrote:
[long winded approach to the actual question snipped]
Now, it seems to me that since we've got something in the tensor
language of general relativity, we should be able to do general
relativity with this. But, obviously, we can't, or people would be
doing this. Why don't we do this (or do we, and no one has told me
about it)? At what point does this break and not make any sense
anymore? Can we at least do relativistic quantum mechanics like this
if we wanted to?

Um. Do you mean, why has nobody produced a quantum mechanical
theory corresponding to general relativity?

They have. The problem is, there are infinities that cannot be
gotten rid of in the usual fashion. The buzz phrase is, GR is
not renormalizable.

Capsule explanation in words, with *plenty* of huge holes in
it follows. If you want to actually learn this, you get to
suffer through a master's degree in it, same as everybody else.

Consider how quantum mechanics describes an interaction. There's
a line coming along representing an electron, and a wavy line
for a photon. The photon and the elctron hit, you get this new
thing that is like an electron but in a strange state called
"virtual." It goes along for a while, then spits a photon out
at some new energy and angle, and goes on by itself. The strength
of this is directly calcuable by quantum electrodynamics (QED)
and comes out pretty spiffy. This is a tiny little bit of the
notion of perturbation theory, which is a large part of what
we know how to do in particle physics.

People imagined gravity waves replacing the photon, and went ahead
and calculated mass/gravity wave interactions. So far so good.
The coupling constant is minute though, and it's very hard to
do experiments equivalent to, say, a radio antenna. So far, nobody
has managed to detect a gravity wave. (At least, not directly
and not with any reliability. Some tantalizing hints that have
been yakked about endlessly, but left in the "unsure" pile.)

Now here comes the inifinities. An electron carries with it an
electric field. This field consists (in QED) of photons that
it emits then re-absorbs. You get little loops. And each one
of those loops produces an infinity. Similarly, that interaction
I mentioned can involve an extra photon being emitted and then
re-absorbed, changing the vertex into a loop. Each loop involves
integrating over the energy of the photon, and that makes the
loop infinite.

In QED, this is gotten around by renormalizing. It is pointed
out that we do not observe the actual charge and mass of the
electron, but the net result after all the loops. So, through
various mathematical methods (look up the term "regulator")
we adjust the bare mass and charge so that the observed value
comes out right. In a renormalizable theory, it can be shown
that we only need a small number of parameters to adjust, and
it will make all interactions come out finite. In QED we need
the self energy (the mass) and the interaction (the charge).
Once we have those (say by two experiments to measure them)
we can predict all other experiments. And it's all very spiffy.
By adjusting just those few parameters, we get rid of all of
the infinities.

In GR, the problem is, you can't get rid of all the infinities
with just a few parameters. Each time we add a loop, it's a new
and completely independent infinity. Adjusting a parameter to
fix that infinity means we do nothing for the next loop. Another
buzz phrase is "primitive divergence." GR has an arbitrary number
of them. So, we would need a free parameter for each new
interaction we wanted to test. That means that quantum GR has
essentially no predictive ability, as each interaction has a
free parameter to tune to adjust it to come out right.

Still, people have done some interesting work with quantum GR.
There are some cool things you can pull out from various
theorms and conditions, especially based on exact solutions
to the full GR equations. There was some cool work on quantum
effects at the horizon of a black hole, for example. And there
has been quite a bit of work on quantum mechanics on a
background of classical general relativity.
Socks

cfgauss wrote:
I have been studying tensor calculus in relativity a little bit before
school starts, and I have a question that I haven't been able to answer
by myself. I know that we can rewrite Maxwell's equations using the
electromagnetic field tensor,
F^(mu nu) =
(0 E^1 E^2 E^3)
(-E^1 0 B^3 -B^2)
(-E^2 -B^3 0 B^1)
(-E^3 B^2 -B^1 0)
And we can re-write Maxwell's equations as, with d meaning curly-d, and
J is the 4-current
d_nu F^(mu nu) = 4 pi J^mu
d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) = 0

With which we can do all kinds of nice special relativity-stuff, and, I
presume, general relativity stuff, too (although I haven't gotten that
far in what I've been studying).

Now, I know that in quantum mechanics, we have a complex valued state
vector Psi, satisfying some differential equation (which we'll assume
is not the Schrödinger equation, but is relativistically correct,
because I've been thinking about relativity). We can write Psi in
terms of real and imaginary parts, say Psi = A + i B. Now, presumably
if we wanted we could re-write the differential equation that Psi
satisfies as two separate differential equations, one in terms of A and
one in terms of B. Now, it seems like we could form another "quantum
mechanical field tensor" in terms of the components of A and B, just
like we did for E and B, and re-write our differential equations as
tensor equations like we did before, and do more relativity-stuff.

You may be confusing the "state vectors" of quantum mechanics with
vectors in R^3, i.e. 3-dimensional space. A vector in R^3 can be
expressed as the sum of three terms, each of which is a scalar times
one of the three basis vectors:
A = A_x i + A_y j + A_z k
Here, i, j, and k are the unit vectors in the x, y, and z directions.
Together they form a set of basis vectors, or basis, for vectors in
R^3.

However, to express the state vectors of quantum mechanics, we often
need an infinite number of basis vectors. For example, for a system in
which the only dynamical variable is the position of a particle, we
would have a basis vector |x,y,z for each position the particle might
be at, and the state vector could be expressed as
int Psi(x,y,z) |x,y,z dx dy dz
where Psi(x,y,z) is a complex-valued function called the wavefunction.

For multiple particles it gets even more complicated. For a system in
which we had two particles, each with a different position, we would
have a basis vector for each possible set of positions, such as
|x1,y1,z1,x2,y2,z2,
and the wavefunction would be a function of six variables:
Psi(x1,y1,z1,x2,y2,z2)

On the other hand, there are some variables in quantum mechanics that
take on only a finite number of discrete states, such as the spin
angular momentum of the electron. If we were only concerned with the
spin of an electron, we could consider only two basis vectors, one in
which the spin was up, and one in which the spin was down. With these
state vectors, you can play games somewhat similar to the one you're
suggesting. For bosons (integer-spin particles), you can get vectors
and tensors out of this. But for fermions (integer-and-a-half-spin
particles), such as electrons, you end up with a new kind of beast
called a spinor.

cfgauss wrote:
I have been studying tensor calculus in relativity a little bit before
school starts, and I have a question that I haven't been able to answer
by myself. I know that we can rewrite Maxwell's equations using the
electromagnetic field tensor,
F^(mu nu) =
(0 E^1 E^2 E^3)
(-E^1 0 B^3 -B^2)
(-E^2 -B^3 0 B^1)
(-E^3 B^2 -B^1 0)
And we can re-write Maxwell's equations as, with d meaning curly-d, and
J is the 4-current
d_nu F^(mu nu) = 4 pi J^mu
d_lambda F_(mu nu) + d_mu F_(nu lambda) + d_nu F_(lambda mu) = 0

With which we can do all kinds of nice special relativity-stuff, and, I
presume, general relativity stuff, too (although I haven't gotten that
far in what I've been studying).

Now, I know that in quantum mechanics, we have a complex valued state
vector Psi, satisfying some differential equation (which we'll assume
is not the Schrödinger equation, but is relativistically correct,
because I've been thinking about relativity). We can write Psi in
terms of real and imaginary parts, say Psi = A + i B. Now, presumably
if we wanted we could re-write the differential equation that Psi
satisfies as two separate differential equations, one in terms of A and
one in terms of B. Now, it seems like we could form another "quantum
mechanical field tensor" in terms of the components of A and B, just
like we did for E and B, and re-write our differential equations as
tensor equations like we did before, and do more relativity-stuff.
Now, it seems to me that since we've got something in the tensor
language of general relativity, we should be able to do general
relativity with this. But, obviously, we can't, or people would be
doing this. Why don't we do this (or do we, and no one has told me
about it)? At what point does this break and not make any sense
anymore? Can we at least do relativistic quantum mechanics like this
if we wanted to?

No, because particle physics cannot be described only in terms of
geometry without getting into problems of infinity, as it is the case
with GR where gravity is the effect of the geometry of spacetime.

QM: G = 0, h = h
GR: G = G, h = 0

Purely incompatible and eventually either one or both must be rejected.

Mike





I'd really appreciate any comments anyone has about this!

Thanks,
Jeremy Price

cfgauss wrote:
With which we can do all kinds of nice special relativity-stuff, and, I
presume, general relativity stuff, too (although I haven't gotten that
far in what I've been studying).

Now, I know that in quantum mechanics, we have a complex valued state
vector Psi, satisfying some differential equation (which we'll assume
is not the Schrödinger equation[...]

Everything is still there.

The Maxwell equations hold regardless of whether it's classical or
quantum theory. The only novel feature is that the field components
belong to a non-commutative number system in quantum theory. But they
satisfy the exact same differential equations.

You have states regardless of whether it's classical or quantum theory.
All a state is is a functional that acts on variables to yield numeric
"expectation" values. These are present in both classical and quantum
theory. What you're calling the "Psi" are a special instance of the
more general concept of "pure" state, which is generic to all physics,
classical or quantum. Corresponding to the vector |psi is the
valuation operator A |- psi| A^ |psi (where A^ is the representation
of A^ in the vector space spanned by A); similarly, corresponding to a
classical pure state is a valuation operator of the form A |- f(A)
satisfying the properties:
f(A+B) = f(A) + f(B)
A = 0 -- f(A) = 0
f(1) = 1
f(AB) = f(A) f(B).
Only the last property distinguishes classical state from quantum
states; and classical pure states from classical mixed states.

There are also the more general concepts of mixed states which are
generic to classical and quantum physics; mixed states do NOT take on
the form of |psi's, but rather the form
W = sum p_i |psi_ipsi_i|
p_1 + ... + p_n = 1; p_i = 0
with the corresponding valuation
W[A] = sum p_i psi_i| A |psi_i.
In fact, its for mixed states and ONLY mixed states that you have any
notion of probability (in BOTH classical or quantum physics); the
coefficients p_1, ..., p_n being naturally interpreted as
probabilities. Pure states do not have any notion of probability
associated with them. A probability interpretation (in either
classical or quantum physics) can only come about by posing some
mechanism for converting a pure state to a mixed state; e.g.
sum c_i |psi_i -- [conversion] -- sum |c_i|^2 |psi_ipsi_i|,
which is NOT mandated by anything in the theory, per se!

Indeed, the biggest crime of your typical textbook presentation (and of
the freshman view of the subject engendered by it) is confusing
superpositions with mixed states and attributing all the explanations,
visualizations, etc. of the latter to the former which it has nothing
to do with. Nothing is a superposition, per se; since every pure state
can be written (with respect to a suitable basis) as |psi = 1 |0.
Everything is a superposition, including things that "are not", since
every pure state can be written (with respect to a suitable basis) in
ANY arbitrary form |psi = sum p_i |psi_i; i = 1,...,n for ANY
(p_1,...,p_n) and n = 1. So calling something a "superposition" means
and says nothing. It's actually mixed states people are talking about
(even when they don't realize it), when they're bringing up the
concepts of states somehow being composed probabilistically of other
states.

Your equation quantum=state; classical=variable is too narrow and is
largely a by-product of the narrowness of the typical textbook
presentation. All 4 combinations are present; quantum=state,
classical=state, quantum=variable, classical=variable.

Similarly; all combinations exist: quantum & Hilbert space; classical &
Hilbert space (i.e. a Hilbert space with 1-dimensional superselection
sectors) as well as the intermediary: hybrid quantum-classical &
Hilbert space (Hilbert space with more than 1 superselection sector;
and at least one superselection sector of dimension greater than 1)...

.... and quantum & phase space (for Weyl quantization, the phase space
density is not positive definite, which limits the range of sets which
have positive measure with respect to the density; Berezin
quantization, on the other hand, works directly off the classical phase
space, so doesn't have this issue; and is related to Weyl quantization
by a Gaussian smearing); and classical & phase space ... as well as
intermediaries of hybrid classical-quantum & phase space.

Mike wrote:
No, because particle physics cannot be described only in terms of
geometry without getting into problems of infinity, as it is the case
with GR where gravity is the effect of the geometry of spacetime.

QM: G = 0, h = h
GR: G = G, h = 0

That's

QM: G = 0, h = h, c = infinity
G(alilean)GR: G = G, h = 0, c = infinity

QFT: G = 0, h = h, c = c
P(oincare')GR: G = G, h = 0, c = c

Purely incompatible and eventually either one or both must be rejected.

There is no incompatibility for h = h vs. h = 0, for c = infinity; and
the question of incompatibility for c = c is open and (given my
foregoing remarks here and elsewhere over the past several months),
almost certainly not present either.

In any case, the issue and clear culprit, given the above list, is c =
c (dovetailing right into the foregoing remarks mentioned above).

Puppet_Sock wrote:
Capsule explanation in words, with *plenty* of huge holes in
it follows. If you want to actually learn this, you get to
suffer through a master's degree in it, same as everybody else.

Consider how quantum mechanics describes an interaction. There's
a line coming along representing an electron, and a wavy line
for a photon. The photon and the elctron hit, you get this new
thing that is like an electron but in a strange state called
"virtual." It goes along for a while, then spits a photon out
at some new energy and angle, and goes on by itself. The strength
of this is directly calcuable by quantum electrodynamics (QED)
and comes out pretty spiffy. This is a tiny little bit of the
notion of perturbation theory, which is a large part of what
we know how to do in particle physics.

People imagined gravity waves replacing the photon, and went ahead
and calculated mass/gravity wave interactions. So far so good.
The coupling constant is minute though, and it's very hard to
do experiments equivalent to, say, a radio antenna. So far, nobody
has managed to detect a gravity wave. (At least, not directly
and not with any reliability. Some tantalizing hints that have
been yakked about endlessly, but left in the "unsure" pile.)

Now here comes the inifinities. An electron carries with it an
electric field. This field consists (in QED) of photons that
it emits then re-absorbs. You get little loops. And each one
of those loops produces an infinity. Similarly, that interaction
I mentioned can involve an extra photon being emitted and then
re-absorbed, changing the vertex into a loop. Each loop involves
integrating over the energy of the photon, and that makes the
loop infinite.

In QED, this is gotten around by renormalizing. It is pointed
out that we do not observe the actual charge and mass of the
electron, but the net result after all the loops. So, through
various mathematical methods (look up the term "regulator")
we adjust the bare mass and charge so that the observed value
comes out right. In a renormalizable theory, it can be shown
that we only need a small number of parameters to adjust, and
it will make all interactions come out finite. In QED we need
the self energy (the mass) and the interaction (the charge).
Once we have those (say by two experiments to measure them)
we can predict all other experiments. And it's all very spiffy.
By adjusting just those few parameters, we get rid of all of
the infinities.

In GR, the problem is, you can't get rid of all the infinities
with just a few parameters. Each time we add a loop, it's a new
and completely independent infinity. Adjusting a parameter to
fix that infinity means we do nothing for the next loop. Another
buzz phrase is "primitive divergence." GR has an arbitrary number
of them. So, we would need a free parameter for each new
interaction we wanted to test. That means that quantum GR has
essentially no predictive ability, as each interaction has a
free parameter to tune to adjust it to come out right.

Continuing in the same style:

The basic difference between QED and this approach to quantum gravity is
that in QED the field quanta (photons) are neutral, so there are only
loops on the charged particle lines. In GR, however, the "charge" is
energy, and that is also carried by the field quanta. So instead of just
loops on the particle lines, you get loops on the quanta lines as well,
and loops on the loops, and loops on the loops on the loops.... The
divergence is of a whole different order than in QED.


Tom Roberts

"Puppet_Sock" wrote in message
oups.com...
cfgauss wrote:
[long winded approach to the actual question snipped]
Now, it seems to me that since we've got something in the tensor
language of general relativity, we should be able to do general
relativity with this. But, obviously, we can't, or people would be
doing this. Why don't we do this (or do we, and no one has told me
about it)? At what point does this break and not make any sense
anymore? Can we at least do relativistic quantum mechanics like this
if we wanted to?

Um. Do you mean, why has nobody produced a quantum mechanical
theory corresponding to general relativity?

They have. The problem is, there are infinities that cannot be
gotten rid of in the usual fashion. The buzz phrase is, GR is
not renormalizable.

Capsule explanation in words, with *plenty* of huge holes in
it follows. If you want to actually learn this, you get to
suffer through a master's degree in it, same as everybody else.

Consider how quantum mechanics describes an interaction. There's
a line coming along representing an electron, and a wavy line
for a photon. The photon and the elctron hit, you get this new
thing that is like an electron but in a strange state called
"virtual." It goes along for a while, then spits a photon out
at some new energy and angle, and goes on by itself. The strength
of this is directly calcuable by quantum electrodynamics (QED)
and comes out pretty spiffy. This is a tiny little bit of the
notion of perturbation theory, which is a large part of what
we know how to do in particle physics.

People imagined gravity waves replacing the photon, and went ahead
and calculated mass/gravity wave interactions. So far so good.
The coupling constant is minute though, and it's very hard to
do experiments equivalent to, say, a radio antenna. So far, nobody
has managed to detect a gravity wave. (At least, not directly
and not with any reliability. Some tantalizing hints that have
been yakked about endlessly, but left in the "unsure" pile.)

Now here comes the inifinities. An electron carries with it an
electric field. This field consists (in QED) of photons that
it emits then re-absorbs. You get little loops. And each one
of those loops produces an infinity. Similarly, that interaction
I mentioned can involve an extra photon being emitted and then
re-absorbed, changing the vertex into a loop. Each loop involves
integrating over the energy of the photon, and that makes the
loop infinite.

In QED, this is gotten around by renormalizing. It is pointed
out that we do not observe the actual charge and mass of the
electron, but the net result after all the loops. So, through
various mathematical methods (look up the term "regulator")
we adjust the bare mass and charge so that the observed value
comes out right. In a renormalizable theory, it can be shown
that we only need a small number of parameters to adjust, and
it will make all interactions come out finite. In QED we need
the self energy (the mass) and the interaction (the charge).
Once we have those (say by two experiments to measure them)
we can predict all other experiments. And it's all very spiffy.
By adjusting just those few parameters, we get rid of all of
the infinities.

Another way of looking at is explained in Zee's excellent book on Quantum
Field Theory - Quantum Field Theory in a Nutshell. The modern way of
looking at it to impose a cutoff (technically called a regularization - we
can cutoff energy and even - wow - dimensions). The bottom line is once we
impose this cutoff in both gravity and EM the infinites disappear. For
gravity you may find the following interesting
http://arxiv.org/abs/gr-qc/9512024.
But getting back to what I was saying when we impose the cutoff in EM we
find something quite interesting - we can rewrite the equations in terms of
physically measurable quantities called the renormalized mass and charge.
This is the trick of renormalization - physically measurable quantities
depend on the cutoff. Intuitively what is going on for charge is we have
all these virtual photons around a charge that screen it like a dielectric.
As we get closer and closer to the charge (ie we measure it using higher
energies) the screening effect becomes less so its measured value is larger.
All renormalization says is we should use the measured value in our
calculations. But the thing to note is to make physical sense of the theory
we still need to impose a cutoff - if you do not you run into what is called
the landau pole. But since we know long before that another theory takes
over - the electroweak theory - it is not really a problem. Of course that
theory has it own version of the landau pole but what is thought is that a
theory combining all the fundamental forces (except gravity) may not have
this problem. Unfortunately the same trick does not work for gravity - it
is not renormaliseable. But what is thought is that a theory combining
gravity and all the other forces will not have this problem and we can
calculate things to all energies and not have to impose this annoying
cutoff.

Thanks
Bill


In GR, the problem is, you can't get rid of all the infinities
with just a few parameters. Each time we add a loop, it's a new
and completely independent infinity. Adjusting a parameter to
fix that infinity means we do nothing for the next loop. Another
buzz phrase is "primitive divergence." GR has an arbitrary number
of them. So, we would need a free parameter for each new
interaction we wanted to test. That means that quantum GR has
essentially no predictive ability, as each interaction has a
free parameter to tune to adjust it to come out right.

Still, people have done some interesting work with quantum GR.
There are some cool things you can pull out from various
theorms and conditions, especially based on exact solutions
to the full GR equations. There was some cool work on quantum
effects at the horizon of a black hole, for example. And there
has been quite a bit of work on quantum mechanics on a
background of classical general relativity.
Socks

Bill Hobba wrote:
Another way of looking at is explained in Zee's excellent book on Quantum
Field Theory - Quantum Field Theory in a Nutshell. The modern way of
looking at it to impose a cutoff (technically called a regularization - we
can cutoff energy and even - wow - dimensions). The bottom line is once we
impose this cutoff in both gravity and EM the infinites disappear. [...]
when we impose the cutoff in EM we
find something quite interesting - we can rewrite the equations in terms of
physically measurable quantities called the renormalized mass and charge.
[...]

Right. But an important fact you forgot to mention is that the results
do not depend on the choice of cutoff. The ratios of renormalized/bare
quantities do depend on the cutoff, but as long as you use measured
values for the renormalized mass and renormalized charge the dependency
on cutoff disappears (as long as it is high enough).


Tom Roberts

wrote:
Mike wrote:
No, because particle physics cannot be described only in terms of
geometry without getting into problems of infinity, as it is the case
with GR where gravity is the effect of the geometry of spacetime.

QM: G = 0, h = h
GR: G = G, h = 0

That's

QM: G = 0, h = h, c = infinity
G(alilean)GR: G = G, h = 0, c = infinity


what?



QFT: G = 0, h = h, c = c
P(oincare')GR: G = G, h = 0, c = c

what?



Purely incompatible and eventually either one or both must be rejected.

There is no incompatibility for h = h vs. h = 0, for c = infinity; and
the question of incompatibility for c = c is open and (given my
foregoing remarks here and elsewhere over the past several months),
almost certainly not present either.

We see you in Sweden then soon if you say so. or get real.

Mike


In any case, the issue and clear culprit, given the above list, is c =
c (dovetailing right into the foregoing remarks mentioned above).