Quantum Field Theory: The Big, Simple Picture?
Jay R. Yablon
and for all the drama and encyclopedic tomes about quantum theory, it
seems to me that there is really a very simple and clear line between
classical and quantum theory, at least when one views QFT through the
lens of Path Integration.
For classical theory, one has a field equation, or "equation of
motion," such as, but not limited to, Maxwell's
J^u=d_vF^vu (1)
or Einstein's
-kT^uv=R^uv-.5g^uvR (2)
One then obtains a Lagrangian density L which replicates the classical
field equation through the Euler-Lagrange prescription. In the former
case, we have the Maxwell Lagrangian density and in the latter the
Einstein-Hilbert Lagrangian where L_matter is a constant (I think -.5)
times the trace T=T^s_s.
For QFT, setting aside the fact that some path integrals are very
hard to calculate exactly, the conceptual prescription is really very
simple: Take L for the field psi, and put it in the path integral:
Z = $dpsi exp i[d^4x sqrt(-g) L] (3)
Then, calculate the intergal. Period.
When one does the calculation, and if one can do the calculation exactly
and preferably keep the boundary terms too, we then obtain an equation
for transition amplitudes which tells what our classical field theory
turns into once it is "path integral" quantized.
Yes, there is a lot of detail which emerges from all of this, and
sometimes calculating (3) exactly to all orders and leaving out nothing
as an approximation or simplification can be intractably difficult as a
mathematical challenge (and you have to add a few things along the way
such as +i epsilon to avoid poles when doing an inverse Fourier
transformation of the propagator and you may run into renormalization
problems). But fundamentally, conceptually, it seems to me that that is
it, and that the boundary between and the transition from classical to
quantum theory is just that simple, and that everything else is doing
calculation and overcoming obstacles in doing the calculations. I know
that I am totally ignoring canonical quantization, though from what I
understand, that and path integral quantization are two alternative
approaches (and the only two known approaches), so if you have one, you
implicitly have both. Am I being naive or missing something big?
Thanks,
Jay
____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm
X-Phy
> � � I have been studying quantum field theory for about two years now,
> and for all the drama and encyclopedic tomes about quantum theory, it
> seems to me that there is really a very simple and clear line between
> classical and quantum theory, at least when one views QFT through the
> lens of Path Integration.
But that isn't a line. That is a formal transformation that, from a
classical theory, leads to the corresponding quantum mechanical one.
Be it a field equation, a material point in a potential well, a
string, a brane, a fluid... This transformation is aptly called a
quantization.
> � � For classical theory, one has a field equation, or "equation of
> motion," such as, but not limited to, Maxwell's
>
> J^u=d_vF^vu (1)
>
> or Einstein's
>
> -kT^uv=R^uv-.5g^uvR �(2)
>
> One then obtains a Lagrangian density L which replicates the classical
> field equation through the Euler-Lagrange prescription. �In the former
> case, we have the Maxwell Lagrangian density and in the latter the
> Einstein-Hilbert Lagrangian where L_matter is a constant (I think -.5)
> times the trace T=T^s_s.
>
> � � For QFT, setting aside the fact that some path integrals are very
> hard to calculate exactly, the conceptual prescription is really very
> simple: �Take L for the field psi, and put it in the path integral:
>
> Z = $dpsi exp i[d^4x sqrt(-g) L] � (3)
>
> Then, calculate the intergal. �Period.
>
> When one does the calculation, and if one can do the calculation exactly
> and preferably keep the boundary terms too, we then obtain an equation
> for transition amplitudes which tells what our classical field theory
> turns into once it is "path integral" quantized.
That's it.
> � � Yes, there is a lot of detail which emerges from all of this, and
> sometimes calculating (3) exactly to all orders and leaving out nothing
> as an approximation or simplification can be intractably difficult as a
> mathematical challenge (and you have to add a few things along the way
> such as +i epsilon to avoid poles when doing an inverse Fourier
> transformation of the propagator and you may run into renormalization
> problems).
They are not only intractable, but sometimes difficult to justify at
the fondamental, theoretical level, such as renormalization. But
actually, it is nothing but quantization. Quantum Filed Theory =
quantization of Classical Field Theory, that is, second quantization.
> But fundamentally, conceptually, it seems to me that that is
> it, and that the boundary between and the transition from classical to
> quantum theory is just that simple, and that everything else is doing
> calculation and overcoming obstacles in doing the calculations. �I know
> that I am totally ignoring canonical quantization, though from what I
> understand, that and path integral quantization are two alternative
> approaches (and the only two known approaches), so if you have one, you
> implicitly have both. �Am I being naive or missing something big?
The paths integral formalism works only in the cases where a
Lagrangian can be defined. There are sytems for which it can't,
called non holonomic systems. So it is less fundamental than the
canonical quantization. At the origin, it was thought as an
interpretation of quantum mechanics, but it is only a (not new)
reformulation that brings no insight, since it applies to a classical
field as well.
There are other quantization approaches, such as geometrical
quantization or stochastic quantization. Anyway, starting from a
classical theory is not satisfying, and there are quantum systems
whithout classical counterpart, such as spin 1/2 particles.
--
X-Phy
glird
>
><� I have been studying quantum field theory for about two years now, and for all the drama and encyclopedic tomes about quantum theory, it seems to me that there is really a very simple and clear line between classical and quantum theory, at least when one views QFT through the lens of Path Integration.
You are anything but "naive" even though you ARE missing something
big. Granted that your mathematical ability is superlative, you -- as
all physicists do -- are ignoring the PHYSICAL MEANINGS of each and
every symbol IN your expertly calculated equations. [For instance,
what IS "L_matter"; and what does "transition amplitudes" measure, if
we actually do the calculation "exactly"; i.e. take the calculus all
the way to the limit instead of only infinitesimally close?]
Far more than that, however what you are missing in the "big"
picture, i.e. our understanding of the structure and modes of action
of the universe, is that
our understanding is independent of any mathematics at all.
glird
brad
> � � I have been studying quantum field theory for about two years now,
Have you seen this book? " How is Quantum Field Theory Possible?"
by, Sunny Y. Auyang ISBN 0-19-509345-3
Brad
X-Phy
> � � I have been studying quantum field theory for about two years now,
> and for all the drama and encyclopedic tomes about quantum theory, it
> seems to me that there is really a very simple and clear line between
> classical and quantum theory, at least when one views QFT through the
> lens of Path Integration.
The issue is a bit entangled because of spurious historical inputs.
Formerly, we had the classical theory, which worked extremely well
especially in celestial mechanics. So planets not yet observed could
be predicted. And also electromagnetism. It was like today with the
standard model. But it has be shown this theory is but an
"approximation" of quantum theory. It is not still clear whether it
is really an approximation, so various methods has been proposed to
quantize the classical theory, some of them having also the ambition
to provide an interpretation. Canonical quantization and paths
integral belong to this category. The latter has a special status,
since a long time ago it was known that the Lagrangian formulism had
strong analogies with the eikonal approximation, that is the
approximation of a wave at high energy. Historically, it has been the
pathway to the first real quantum theory: the Schr�dinger equation.
That gave insight into the classical theory itself, which yet could be
deemed "intuitive" or "natural", and Hamiltonian and Lagragian
formalisms.
So what is the situation now? We have "field" equations, in contrast
to "particles" like the fermions. QFT says that these equations are
the classical counterparts (they not necessary are, like the Dirac
equation), and thus we have a collection of quantum mechanical coupled
harmonic oscillators, one at each point. This description entails the
Bose-Einstein statistics, which is consistent with the bosonic nature
of the fields. Theoretically, there is no problem at this point, but
we have to make calculations in order to verify the QFT hypothesis
through an experimental test. Those calculations could *in principle*
be performed directly from the quantum mechanical formulation, but
because of technical difficulties, it is necessary to revert to a
quantization method. The paths integral, although impaired with its
own technical problems and a dubious interpretation, has historically
been a breakthrough used to derive the Feynman rules of the eponymous
diagrams. Each diagram is a graphical representation of the set of
paths with a common topology.
In conclusion, the real big picture is a quantum mechanical theory in
it own right, which *in principle* suffice to itself. But in pratice,
we must have recourse to old concepts, both in order to understand
what is going on, and to get the numerical figures comparable with the
experimental data. Unfortunately, many things like spin 1/2 and gauge
interactions have no classical counterparts, and we could devote an
entire newsgroup about what is "fundamental", what work or not in such
and such situation, what is "palatable", which limit exist or not...
--
X-Phy