Reconciling quantum theory & relativity (was: QFT Questions)

From: Alfred Einstead (whopkins_at_csd.uwm.edu)
Date: 08/14/04 

Date: 14 Aug 2004 07:58:07 -0400

mhelland@techmocracy.net (Mike Helland) wrote:
> Can anyone clarify some basic things about quantum field theory for me?
>
> Is it true the search for quantum gravity is based on QFT?

More generally, the search to reconcile Quantum Theory with Relativity
is focused on bringing the most developed parts of each (General
Relativity and Quantum Field Theory) onto a common foundation.

> Is it true that in QFT every field is propogating at c?

The propagator of a massless field is a solution to the wave
equation with singular source:
               d^2 u/dt^2 - c^2 del^2 u = k delta(x)
              u(r,0) = du/dt (r,0) = 0

It's non-zero and singular on the light cone; which is the same
thing as saying that the field propagates at light speed.

The propagator for a massive field, a solution to the Klein
Gordon equation
              d^2 u/dt^2 - c^2 del^2 u + m^2 u = k delta(x)
(I might have the signs reversed on the m's and delta's) has
the same singular part on the light cone, plus an additional
non-zero part within the light cone.

> Can you can model quantum mechanics without QFT?
> Would that be non-relativistic quantum mechanics?

More generally, there is no consistent relativistic concept of
a particle. The Heisenberg relations for particle trajectories
t |-> (x(t), y(t), z(t)):
     [x(t), x'(t)] = [y(t), y'(t)] = [z(t), z'(t)] = i h-bar/m
     [x(t),y(t)]=[y(t),z(t)]=[z(t),x(t)] = 0
     [x'(t),y'(t)]=[y'(t),z'(t)]=[z'(t),x'(t)] = 0
     [x(t),y'(t)]=[y(t),z'(t)]=[z(t),x'(t)] = 0
     [y(t),x'(t)]=[z(t),y'(t)]=[x(t),z'(t)] = 0
don't generalize readily for worldlines in Relativity,
           s |-> (t(s), x(s), y(s), z(s)).

More generally, it's not possible to represent particle worldlines
in a relativistic setting in a way compatible with quantum theory
without violating causality. This is a fairly well-known result
explained, for instance, in Ticciati (Quantum Field Theory For
Mathematicians; section 1.6 The Position Operator [and it's
impossibility]).

More generally still, even further than that, there is no concept
of a particle as a corpuscular entity in relativity in the first
place. E = m c^2 means matter can interconvert with energy. What
becomes of a corpuscle that converts, if you assumed that particles
were of this form? In QM, operators apply for all time. So, the
position operator (X(t), Y(t), Z(t)) implicitly assumes that the
thing is around forever in the past and forever into the future.

> What experimental evidence exists for QFT that doesn't exist for
> non-relativistic quantum mechanics?

(1) Particles don't have separate identity as individual corpuscules;
    e.g., there is only 1 way to put two particles of the same type
    into two boxes, one in each; not 2 ways.

In essence, that means all the electrons in the universe is actually
the same electron at different places at the same time. All the
photons is the same photon at different places at the same time.
They have no more identity as individual objects than waves on an
ocean do.

The difference in the way particles are counted affects the
determination of such empirical constants as the specific heats
of materials. And, this discrepancy has been known about since
the early to mid 19th century.

It should also be of interest to remember that in a general
context of spacetime (whether Newtonian or Relativistic, it
doesn't matter) "individual" people don't have existence as
separate individuals either. They don't have worldlines at
all. Rather their "worldlines" are actually strands on a
web-shaped structure that includes everybody on it, including
the food you eat... since you're related to your food. Your
strand is connected to your mother's at that point&time in
spacetime corresponding to where&when you were a fetus.

(2) Particle identity and number are not preserved; and particle
identity and number cannot be construed in any way that enables
one to infer that they are preserved.

Interactions like
                    nu_e + W- --> e

might be interpreted as a number-preserving interaction if one
adopted the fiction that the electron (e) is a bound state
of a neutrino (nu_e) and W-. But then, one also has interactions
like
                W- --> anti-nu_e + e
[which is a prime example of the matter-energy conversion issue
raised above, since the W is a particle of energy] which ruin
that interpretation.

Then you also have interactions such as:
                   Z --> W+ + W-
                   W+ --> Z + W+
                   W+ + W- --> Z
etc.; which completely defy any attempt to categorize as a
number preserving interaction, even modulo any kind of
assumption about their being bound states. For instance,
one has the interaction:
            Z --> W+ + W-
              --> (Z + W+) + W- = Z + (W+ + W-)
              --> Z + Z

which absolutely can never be construed as a particle number
preserving interaction of any kind under any interpretation of
any kind.

> Have attempts been made at quantum gravity without QFT, where
> the graviton is travelling faster than c?

The light cone, itself, is the root of the problem both (1)
in trying to reconcile GR and QFT and (2) in trying to even
formulate a consistent QFT, which is still an open problem.

Since the propagators are singular on the light cone, then
the fields are represented in terms of singular delta-like
functions. But the field equations are non-linear, which
means they involve non-linear combinations of singular
functions ... which is generally ill-defined and lies at
the root of all the problems with the infinities in
quantum fields (and even for classical fields).

String theory is supposed to ameliorate that issue by
modelling fields not in terms of point-like particles, but
in terms of line-like particles.

But that way of resolving (2) can never be anything more
than a workaround, since it doesn't even begin to address
the problem of (1).

The problem of (1) runs much deeper, isn't even recognized
(much less addressed) by either of the main approaches to
quantum gravity (loop QG or string theory) and entails a
resolution much more striking.

Quantum gravity is supposed to quantize the metric for
spacetime itself. That means that the coefficients in
the line element:
             ds^2 = sum (g_{mn} dx^m dx^n: m,n=0,1,2,3)
are no longer even well-defined numbers, but instead have
values which are state-dependent and take on a dispersed
distribution about a "expectation" value.

The light cone, itself, is defined by
                   ds^2 = 0.
Likewise, the inside and outside of the light cone are
defined by ds^2 > 0 and ds^2 < 0.

But the defininition the various objects which enter into
quantum field theory requires one FIRST write down the
commutators
              [A(x), B(x+dx)] = 0 if ds^2 < 0
which assumes that the light cone has ALREADY been defined.
So, in order to define the g_{mn}'s one needs to first have
a definition of ds^2. And in order to define ds^2, one
needs to first have a definition of the g_{mn}'s.
 
That's a catch-22 with no way out.

It means that the causality principle
              [A(x), B(x+dx)] = 0 if ds^2 < 0
can't be consistently maintained.

At root of this problem is that the light cone, being defined
in terms of the g's (which are now smeared out with non-zero
dispersions), is smeared out and is no longer a sharply defined
object. The actual location of the light cone is state-dependent.
So, what looks like a causal interaction in one state may be
causality violating seen in another state.

The punchline, of course, is that states can combine by
superposition. So, you can have a superposition of states,
where the interaction is both causal and causality-violating.

This type of anomaly doesn't have a concept or name, so I
called it "Light Cone Tunnelling". Using the loophole provided
by the non-sharpness of the light cone in the context of a
quantized metric, you can actually tunnel out from under the
light cone.

There is a close analogue to this, however, that is already
well-known. The event horizon of a black hole is actually
part of the light cone. It's stationary because all along
the event horizon, the outward radial speed of light is 0.

In this case, light cone tunelling corresponds to energy
that comes out from the black hole. This is already known
as Hawking Radiation.

The way it is explained in more prosaic terms in terms of
ordinary QFT makes essential use of the ambiguity of the
vacuum state.

In general relativistic context, it is actually the case
that the distinction between:
      (particle A present) <-> (anti-particle A absent)
      (particle A absent) <-> (anti-particle A present)
is observer-dependent. Two observers in different states
of acceleration see things differently. This means,
among other things, that what counts as a vacuum in one
frame of reference is not so in the other, since the
absence of all A's in one frame will be seen as a
plenum of anti-A's in the other frame.

So, in a strong gravitational field, it's possible to
mine the vacuum and steal away a particle which, from
afar, looks like one is absorbing the absence of the
opposite particle (i.e., absorbing negative energy).
>>From afar, the black hole takes in negative energy, which
is just a fancy way of saying that it's spitting out positive
energy -- the Hawking radiation.

This is the mechanism by which energy tunnels through the
light cone, in this specific context.

In a more general context, this should also provide the
means by which energy can tunnel through light cones and
produce a faint shadow of causality violation.

So, the "striking" consequence I alluded to above is
causality violation, itself. The roadblock that keeps
quantum theory from being reconciled with relativity
is the unwillingness to drop the assumption of causality.
That's the root of all the problems of bringing the two
theories together.

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