What is gauge
invariance, by the way? Well, here on
Sunday night, January 6th of 2013, let’s not precisely answer that
question just yet. Instead, I would like
to extract some quotes from “A Dictionary of Physics,” published by good ol’
Oxford University Press in 1996.
There is no entry for
gauge invariance, but there is one
for gauge boson, and one for gauge theory.
Last paragraph of the
gauge theory entry—well, the 2nd paragraph of two—is:
“In gauge theories the
interactions between particles can be explained by the exchange of particles
(intermediate vector bosons, or gauge bosons), such as gluons, photons, and W
and Z bosons.”
Which leads us back to
the entry for gauge bosons: “A spin-one
vector boson that mediates interactions governed by gauge theories. Examples of gauge bosons are photons, in
q.e.d., gluons in q.c.d. and W and Z bosons that mediate the interactions in
the Weinberg-Salam model (see electro-weak theory) unifying electromagnetic and
weak interactions. If the gauge symmetry
of the theory is unbroken, the gauge boson is massless. Examples of massless gauge bosons include the
photon and gluon.
“If the gauge symmetry
of the theory is a broken symmetry, the gauge boson has a non-zero mass,
examples being the W and Z bosons.
Treating gravity, as described by the general theory of relativity, as a
gauge theory, the gauge boson is the massless spin-two graviton.”
I don’t want to treat
gravity as a gauge theory! So how to do
it? Goodnight.
Well, first I want to write
down the dictionary’s definitions of broken symmetry, electroweak theory,
q.e.d. and q.c.d. The initials are my
abbreviations, not the dictionary’s, on previous page as well as on this one. Now maybe I can say goodnight and mean
it! 9:45 p.m.
5 a.m. almost, Thurs.
Jan. 10th 2013. So, if the
e.m. field could be “turned off,” isotopic spin symmetry would be exact. This is stated in ‘t Hooft’s Sci. Am. article
of June 1980, first page.
If you are wanting to
have unified description of forces, it would be best to have the isotopic spin
be exact when the e.m. field is turned ON, I would think.
“In order to make a
theory invariant with respect to a local transformation, something new must be
added: a force.” P. 5 of ‘t Hooft article. Need to look at Holstein book to compare how
he discusses this.
We’ve got: force = field = particle (virtual). Not good!
How to make it better?
See also Icke’s
description in his Force of Symmetry book of how uncertainty prinziple governs
virtual particle so-called interaction.
Now, Icke’s book is pretty Ickey, but I do want to keep his virtual
particle description in mind. Gerard ‘t
Hooft writes well. Can’t say same for his
fellow Dutchman, Vincent Icke.
Later.
All right, 8:45 pm Jan
10th Thurs., in bed now, with Jessie and Icke. And a pen, this journal, and a desire to poke
holes in the modern description of symmetry.
Which is why I’ve
returned to Icke’s book. Chapter 7,
“Symmetries,” to be precise. “There must
be at least some things that are more allowed than others.” = end of 1st
paragraph.
2nd
paragraph talks about many things being forbidden, “and it is the forbidding
rules that give structure to the world.”
Who boy, that’s a compelling thought!
But summarizing the “thou shalt not” rules is overwhelming.
So we want to “lump
together a large (possibly infinite) number of rules in a single
all-encompassing one. Thus you might
write in the Constitution of your universe, “Do unto others as thou wouldst
others do unto thee.” Such an overall
rule, Icke says, “that summarizes a multitude of individual possibilities, is
called a symmetry.”
Redundancy is what it’s
coming from, however. Think about that,
and realize you don’t want redundancies to rule the basic theory of …
everything!
Sleepy now. Went to Lake Village today, for SEAEDD
meeting. Goodnight.
10:30 a.m. Sunday
January the 13th 2013.
How would I explain
“gauge invariance” if asked? (Dvorak
Symphonies 5, 7, 9 playing on CD changer.)
Well, it would be difficult. On
page 112 herein, I note that there was no entry in the Oxford Dictionary of
Physics for gauge invariance, only one for “gauge theories,” and one for “gauge
bosons”.
So let me start trying
to think of how to explain gauge invariance by enumerating other terms in
physics where “gauge” is present: gauge
theory, gauge boson, gauge transformation, gauge condition, pressure gauge – ha
ha ha got you there!
I need to go back to Holstein’s
book. I’ve quoted it in here, I’m sure,
but don’t recall when, and so will turn pages backwards and see what I said
about what he said. Pardon me while I
turn back the pages.
Okay, pages 92 thru
96. Symmetry group, or transformation
group, is the key idea or at least the primary tool used in gauge
theories. After all, “invariance” means
invariance with respect to some transformation.
So “gauge invariance”
means invariance with respect to a gauge transformation.
What is a gauge
transformation? (9th Dvorak
Symphony, “New World,” 1st movement, playing.)
Well, of course we have
the global and the local transformations of “gauge.” So “gauge group” seems to
be the most relevant of the terms involving the word “gauge.” In fact, saying the more meaningfully, we
would say “gauge transformation group.”
We are talking about
“potentials” also. Strange word, but
quite relevant! The fields, say E &
B, are invariant with respect to CERTAIN changes in the potentials, Aμ , in
electromagnetism. Now a new term comes
in: “gauge field.” For instance,
Holstein says, “the price for achieving local U(1) invariance is the
introduction of a new gauge field, Aμ
.
So then this means
we’re calling the potential(s) Aμ
a field?
And don’t forget about
the most excellent term “Lagrangian,” used as a noun. Holstein: “… by demanding
the mathematical requirement of local U(1) invariance, we have produced
a Lagrangian that is known to have physical significance.”
To say it another way:
By requiring our Lagrangian to be invariant with respect to a certain gauge
transformation group, U(1), we find that it can be used to generate the Maxwell
equations for the electromagnetic sources and fields.
Of course, saying something is a “member” of a
group is more proper, so leaving out the word “group” above is more proper,
too. But never mind PROPER!
P-ROPER! We gotta get beyond
proper, elegant, and sophisticated. “We”
meaning I. Abolish virtual particlism!
Now it’s about 11:45
a.m. and I’d like to finish this up by noon and get out to the BB Trail with
knee boots on, to take Jessie and myself for a walk. (Steely Dan’s “Black Cow” is playing now.)
The thing to be
discussed is electric charge conservation.
Gerard ‘t Hooft notes that although the absolute value of the phase of
the electron wave in quantum electrodynamics is irrelevant for the outcome of
experiments, “in constructing a theory of electrons it is still necessary to
specify the phase. The choice of a
particular value is called a gauge convention.”
Well now it’s
12:10. Time for a break. Wigner’s Symmetries
and Reflections book, p. 10 on the subject of charge conservation, will be
up next.
