Re: Several questions about Quantum Mechanics

On Apr 8, 8:15 am, ZHANG Pu <Lefthanded.Mater...@xxxxxxxxx> wrote:
4, Is there a widely accepted quantization program of electromagnetic
field? Can you recommend some readings about that?

The free non-interacting electromagnetic field has a well-established
quantization as a system that effectively equates to an infinite
system of simple harmonic oscillators.

In contrast, there is no known quantization for the electromagnetic
field when coupled to other fields; nor (to get technical) even for
the free electromagnetic field in the modern context of electroweak
theory. That is, more generally, the question of how to quantize a
*free* non-Abelian Yang-Mills field is still open (and, essentially,
the subject of a $1000000 prize by the Clay foundation). This devolves
on the electromagnetic field which, in electroweak theory, is mixed in
with the (non-Abelian) field associated with the SU(2) part of the
Standard Model.

To put it slightly differently, the modern treatment of
electromagnetism (since the advent of electroweak theory) replaces the
linear Maxwell equations by non-linear equations that are
inhomogeneous both for the electric and magnetic equations. This is
generally the case for non-Abelian Yang-Mills fields. Quantum field
theory in dimensions greater 2+1 has a difficult time accounting for
any kind of non-linear interactions or self-actions. There is no known
non-trivial quantum field theory in 3+1 dimensions or more.

Resort is made to a power series expansion approach, in the guise of
perturbation theory, for the purpose of trying to account for the non-
linear interactions between fields. The series (that is, the S-matrix
expansion) is not known to converge to a well-defined value, and is
widely believed not to.

Nevertheless, interesting physics arises from what CAN be gleaned. All
of this falls under the header of "renormalization theory" and
involves such concepts as the "renormalization group", the "running of
the couplings", the notions of "bare vs. dressed" charge, etc.

What is NOT as well-known is that what arises shows clearly that the
problem is NOT anything that has per se to do with quantum theory, but
is generic. The various phenomena that are uncovered (particularly the
running of the couplings, which actually manifests in scattering
experiments) are essentially classical in nature. The problem which
plagues quantum theory, which these devices seek to solve, is a
problem that is common to both classical and quantum field theory and
is much older than quantum theory, in fact.

The myth that pervades is that the field divergence problem has to do
with some kind of Planck level physics and is something that can only
be resolved with a theory of quantum gravity. Yet, the phenomena
associated with renormalization theory (particularly the running of
the couplings) are perfectly well visible at present day levels of
resolution -- which is over a dozen orders of magnitude above the
Planck scale. This, alone, serves as a clear hint that we're dealing
with an essentially classical issue, not anything that has to do with
exotic Planck-level or "21st century" Physics.

Indeed, the whole idea (and apparatus) associated with renormalization
theory is much older than quantum theory. It originated with Maxwell,
himself. It was Maxwell who first posed the idea of charge screening,
vacuum polarization, bare vs. dressed charges, setting out to do so
specifically to resolve the problem of field infinities.

The source of the infinity that plagues classical and quantum theory
is easy enough to describe. However, it is difficult to see in the
present-day theoretical literature, because it pertains to an element
that is almost always kept out of the discussion: the vacuum
permittivity.

Consider the equations
div D = rho; D = epsilon E; F = rho E
describing the relation between the D and E fields and the force
density for a charge density rho. Suppose you adopt in place of the
second relation the relation posed originally by Lorentz,
D = epsilon_0 E,
who hypothesized that this relation not only holds in the vacuum but
even in the spaces between the particles that comprise matter. This,
of course, was where the distinction between the "microscopic" vs.
"macroscopic" forms of Maxwell's equations arose.

However, it's wrong and is easy to see. What it actually does is
reintroduce the very field-theoretic infinity that Maxwell went to
great pains to eliminate. It undid the very thesis of the treatise
spelled out in article 61 at the end of chapter 1, and it was not
until the 1940's that the old elements of Maxwell's theory came to be
resurrected (possibly unwittingly) in modern guise as "renormalization
theory".

You can see it as follows: a point like or strongly concentrated
charge distribution for rho, in virtue of the equation (div D = rho)
entails a similarly singular (or near-singular distribution in D).
Given the linear relation (D = epsilon_0 E), this property is
inherited by E, as well. But in order for (rho E) to be well-defined
as a density, E must NOT be singular, wherever rho is.

Check mate.

Hence, the linear relation (D = epsilon_0 E) cannot be valid near
point sources or other strong concentrations of charge. Indeed, this
was precisely the argument set out by Maxwell in Chapter 1 and 2 of
his treatise.

The source of the field infinity is that epsilon was rendered inert
(epsilon_0) and then entirely removed from the discussion ("let
epsilon_0 = 1" or a similar convention). In the process, you lose some
very important physics that it took nearly 50 years after Lorentz to
recover. At of the present day, it has not yet been fully recovered.
There is a more comprehensive account that will cure the field
infinities both classical and quantum theoreitically lurking somewhere
that is properly grounded in classical theory, but it has not yet been
fully developed.

The true nature of the constant can be seen when going over to the
more general context of Yang-Mills theory. Here, the fields (E, B)
become indexed (E^a, B^a) with superscripts, while the charge (e)
becomes indexed as a (Lie-valued) vector e_a. The space of charges
goes from one-dimensional to N-dimensional (e.g. for SU(3) gauge
theory, a classical charge is a 8-dimensional quantity, its quantized
charge occupies a 2-dimensional representation space). In contrast,
the fields (D,H) become indexed with subscripts (D_a, H_a), as the
charge and current.

The relation between the two (D = epsilon E) is no longer a trivial
linear proportionality; nor is it related merely by a duality
transform, as the modern treatment in the language of differential
forms renders the Lorentz relation. The two quantities aren't even of
the same type, so at the conceptual level a linear relation doesn't
even make sense.

Instead, you need to raise and lower indices -- and wherever that
happens, you're talking about a metric of some sort. Here, the metric
is that associated with the underlying symmetry group, and the
relations would be written as D_a = sum (k_ab E^b), where (k_ab) is
the components of this metric. For Maxwell's theory, you don't see the
extra components, so the metric (k) is (mistakenly) forgotten.

In modern theories, these are called Jordan-Brans-Dicke scalars, but
what this actually is is just nothing more than the modern translation
of the permittivity! That is, k_ab <-> epsilon c, and for the dual
metric k^ab <-> mu c.

This metric, in turn, is subject to a dynamics. The dynamics can be
posed by taking the standard geometric formulation of Yang-Mills
fields (principal bundles) and allowing the gauge metric to be
variable. The result is a additional set of equations governing the
dynamics of this set of "dielectric" coefficients.

At the time of Maxwell, epsilon was called K, and Maxwell maMuchde a
very specific point of indicating that this is an entirely non-trivial
element to the theory that cannot be ignored! It is because epsilon
(or Maxwell's K) has been lost that field theory acquired an infinity
at the classical level. In turn, it is because of the problem of the
infinity in the classical theory, that a similar problem came to be
inherited by the quantization of the classical field theories.

Yet, as you can begin to see, the problem is both classical in its
nature -- and in its cure! Quantum gravity and Planck-level Physics is
nothing more than a red herring, and never was anything more than one
here.

Just so there will be no room for questioning Maxwell's (forgotten)
role in the development of all the concepts related to renormalization
theory, I've included the following discussion. Much of this has been
long-forgotten, essentially having been thrown out (like baby with the
bathwater) when the so-called ether theory, which the renormalization
theory has little to do with, had been discarded.

Though it is not well-known today and nearly forgotten, it was Maxwell
who formulated what today is called renormalization theory and the
underlying notion of vacuum polarization

From Article 62 of Maxwell's treatise:
"... the energy of electrification resides in the dielectric medium,
whether that medium be solid, liquid, or gaseous, dense or rare, or
even what is called a vacuum, provided it be still capable of
transmitting electrical action."

"That the energy in any part of the medium is stored up in the form of
a state of constraint called electric polarization, the amount of
which depends on the resultant electromotive intensity at the place."

Which, as per the previous paragraph, includes vacuum polarization.

It is through this theory that the issue of the classical field-
theoretic infinity is resolved (via the notion of charge screening).
The arguments leading to the theory and the resolution of the infinity
transcend the context of the original theory and are not only
applicable across the divide between classical and quantum theory, but
also to the more general context of Yang-Mills theory.

It is of interest to note that the hypothesis that the vacuum behaves
as a dielectric medium not only originates with Maxwell, but is a
central thesis of his entire treatment of classical electromagnetism.

Continuing on:
"That electromotive force acting on a dielectric produces what we have
called electric displacement, the relation between the intensity and
the displacement being in the most general case of a kind to be
afterwards investigated in treating the conduction, but in the most
important cases the displacement is in the same direction as the
intensity, and is numerically equal to the intensity multiplied by K/
(4 pi), where K is the specific inductive capacity of the
dielectric."

In modern language of Yang-Mills theories, K becomes none other than
the gauge metric k_ab, itself, up to a multiple of 4 pi.

From Article 55
"If the electromotive intensity at any point of a dielectric is
gradually increased a limit is at length reached at which there is a
sudden electrical discharge through the dielectric generally
accompanied with light and sound and with a temporary or permanent
rupture of the dielectric."

In the context of article 62, "discharge" is also meant to include the
breakdown of the vacuum itself. It is interesting that this
consequence was never fully spelled out in the treatise. In
particular, what would the products of the dielectric breakdown in a
vacuum be?

As mentioned above, in the theoretical literature, both the vacuum
permeability and vacuum permittivity are disregarded as inert and
inessential, and the constants are never seen, their use being
relegated instead to "engineering applications". Yet the very physics
embodied by the vacuum permittivity is what modern renormalization
theory tries to recapture in the guise of quantum field theory!

This is the source of the problem of the field infinity faced not only
by quantum field theory, but classical theory, as well - after
Lorentz.

The very purpose of Maxwell's hypothesis was to resolve the field
theoretic infinity that would otherwise arise, as stated here:

"the dielectric gives way and its insulating power is destroyed, so
that a current of electricity takes place through it. It is for this
reason that distributions of electricity for which the electromotive
intensity becomes anywhere infinite cannot exist."

This argument on the finiteness of E in the presence of point sources
is quite general and is precisely the argument that is required to
avoid the infinity in the classical theory.

This is how Maxwell avoided the problem that Lorentz reintroduced and
which eventually found its way into quantum field theory.

The modern-day solution partly recovers the old formalism, however,
the resolution is not properly grounded in classical theory and does
not fully embody the Maxwell theory, which requires instead that the
vacuum be treated as a non-trivial dielectric with running
permittivities instead of running couplings.

The notion of charge screening was developed by Maxwell in the
following thought experiment, from Article 55.

"Thus, when a conductor having a sharp point is electrified, the
theory, based on the hypothesis that it retains its charge, leads to
the conclusion that as we approach the point the superficial density
of the electricity increases without limit, so that at the point
itself the surface-density, and therefore the resultant electromotive
intensity, would be infinite. If the air, or other surrounding
dielectric, had an invincible insulating power, this result would
actually occur; but the fact is, that as soon as the resultant
intensity in the neighborhood of the point has reached a certain
limit, the insulating power of the air gives way, so that the air
close to the point becomes a conductor. At a certain distance from the
point the resultant intensity is not sufficient to break through the
insulation of the air, so that the electric current is checked, and
the electricity accumulates in the air round the point."

This was spelled out more clearly in the thought experiment of Article
81 ("A Distribution of Electricity on Lines or Points is Physically
Impossible"):
"If, while [the linear density of charge] remains finite, [the
circumference of a wire] be diminished indefinitely, the intensity at
the surface will be increased indefinitely. Now in every dielectric
there is a limit beyond which the intensity cannot be increased
without a disruptive discharge. Hence a distribution of electricity
in which a finite quantity is placed on a finite portion of a line is
inconsistent with the conditions existing in nature... In the same way
it may be shewn that a point charged with a finite quantity of
electricity cannot exist in nature."

This leads directly to the classical notion of charge screening and
the distinction between bare vs. dressed charges, as spelled out in
Article 83a:
"The apparent charge of electricity within a given region may increase
or diminish without any passage of electricity through the bounding
surface of the region. We must therefore distinguish it from the true
charge, which satisfies the equation of continuity.In a heterogeneous
dielectric in which K varies continuously, if rho' be the apparent
volume-density
del^2 V + 4 pi rho' = 0

Comparing this with [the equation div(K grad V) + 4 pi rho = 0 ], we
find
4 pi (rho - K rho') + grad K . grad V = 0.
The true electrification, indicated by rho, in the dielectric whose
variable inductive capacity is denoted by K , will produce the same
potential at every point as the apparent electrification, denoted by
rho', would produce in a dielectric whose inductive capacity is
everywhere equal to unity."

This, of course, is nothing less than a description back in the 19th
century on the distinction between bare vs. dressed charges.

The distinction is thus grounded firmly at the classical level in the
underlying theory of the universal dielectric medium.

It is thus here that one will find the resolution of the field
theoretic infinity.

.