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John Baez
science forum Guru Wannabe


Joined: 01 May 2005
Posts: 220

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 211) Reply with quote

Also available at http://math.ucr.edu/home/baez/week211.html

March 6, 2005
This Week's Finds in Mathematical Physics - Week 211
John Baez

The last time I wrote an issue of this column, the Huyghens probe was
bringing back cool photos of Titan. Now the European "Mars Express"
probe is bringing back cool photos of Mars!

1) Mars Express website, http://www.esa.int/SPECIALS/Mars_Express/index.html

There are some tantalizing pictures of what might be a "frozen sea" -
water ice covered with dust - near the equator in the Elysium Planitia region:

2) Mars Express sees signs of a "frozen sea",
http://www.esa.int/SPECIALS/Mars_Express/SEMCHPYEM4E_0.html

Ice has already been found at the Martian poles - it's easily visible there,
and Mars Express is getting some amazing closeups of it now - here's a
here's a view of some ice on sand at the north pole:

3) Glacial, volcanic and fluvial activity on Mars: latest images,
http://www.esa.int/SPECIALS/Mars_Express/SEMLF6D3M5E_1.html

What's new is the possibility of large amounts of water in warmer parts of
the planet.

Now for some math. It's always great when two subjects you're interested in
turn out to be bits of the same big picture. That's why I've been really
excited lately about Bott periodicity and the "super-Brauer group".

I wrote about Bott periodicity in "week105", and about the Brauer group
in "week209", but I should remind you about them before putting them together.

Bott periodicity is all about how math and physics in n+8-dimensional space
resemble math and physics in n-dimensional space. It's a weird and wonderful
pattern that you'd never guess without doing some calculations. It shows up
in many guises, which turn out to all be related. The simplest one to verify
is the pattern of Clifford algebras.

You're probably used to the complex numbers, where you throw in just *one*
square root of -1, called i. And maybe you've heard of the quaternions, where
you throw in *two* square roots of -1, called i and j, and demand that they
anticommute:

ij = -ji

This implies that k = ij is another square root of -1. Try it and see!

In the late 1800s, Clifford realized there's no need to stop here. He
invented what we now call the "Clifford algebras" by starting with the
real numbers and throwing in n square roots of -1, all of which anticommute
with each other. The result is closely related to rotations in n+1
dimensions, as I explained in "week82".

I'm not sure who first worked out all the Clifford algebras - perhaps it was
Cartan - but the interesting fact is that they follow a periodic pattern.
If we use C_n to stand for the Clifford algebra generated by n anticommuting
square roots of -1, they go like this:

C_0 R
C_1 C
C_2 H
C_3 H + H
C_4 H(2)
C_5 C(4)
C_6 R(Cool
C_7 R(Cool + R(8)

where

R(n) means n x n real matrices,
C(n) means n x n complex matrices, and
H(n) means n x n quaternionic matrices.

All these become algebras with the usual addition and multiplication of
matrices. Finally, if A is an algebra, A + A consists of pairs of guys
in A, with pairwise addition and multiplication.

What happens next? Well, from then on things sort of "repeat" with period 8:
C_{n+8} consists of 16 x 16 matrices whose entries lie in C_n!

So, you can remember all the Clifford algebras with the help of this
eight-hour clock:


0

R

7 1

R+R C





6 R H 2





C H+H

5 3

H

4


To use this clock, you have to remember to use matrices of the right size to
get C_n to have dimension 2^n. So, when I write "R + R" next to the "7" on
the clock, I don't mean C_7 is really R + R. To get C_7, you have to take
R + R and beef it up until it becomes an algebra of dimension 2^7 = 128. You
do this by taking R(Cool + R(Cool, since this has dimension 8 x 8 + 8 x 8 = 128.

Similarly, to get C_{10}, you note that 10 is 2 modulo 8, so you look at
"2" on the clock and see "H" next to it, meaning the quaternions. But to get
C_{10}, you have to take H and beef it up until it becomes an algebra of
dimension 2^{10} = 1024. You do this by taking H(16), since this has
dimension 4 x 16 x 16 = 1024.

This "beefing up" process is actually quite interesting. For any associative
algebra A, the algebra A(n) consisting of n x n matrices with entries in A
is a lot like A itself. The reason is that they have equivalent categories
of representations!

To see what I mean by this, remember that a "representation" of an algebra
is a way for its elements to act as linear transformations of some vector
space. For example, R(n) acts as linear transformations of R^n by matrix
multiplication, so we say R(n) has a representation on R^n. More generally,
for any algebra A, the algebra A(n) has a representation on A^n.

More generally still, if we have any representation of A on a vector space V,
we get a representation of A(n) on V^n. It's less obvious, but true, that
*every* representation of A(n) comes from a representation of A this way.

In short, just as n x n matrices with entries form an algebra A(n) that's a
beefed-up version of A itself, every representation of A(n) is a beefed-up
version of some representation of A.

Even better, the same sort of thing is true for maps between representations
of A(n). This is what we mean by saying that A(n) and A have equivalent
categories of representations. If you just look at the categories of
representations of these two algebras as abstract categories, there's no
way to tell them apart! We say two algebras are "Morita equivalent" when
this happens.

It's fun to study Morita equivalence classes of algebras - say algebras over
the real numbers, for example. The tensor product of algebras gives us a way
to multiply these classes. If we just consider the invertible classes, we get
a *group*. This is called the "Brauer group" of the real numbers.

The Brauer group of the real numbers is just Z/2, consisting of the classes
[R] and [H]. These correspond to the top and bottom of the Clifford clock!
Part of the reason is that

H tensor H = R(4)

so when we take Morita equivalence classes we get

[H] x [H] = [R]

But, you may wonder where the complex numbers went! Alas, the Morita
equivalence class [C] isn't invertible, so it doesn't live in the Brauer
group. In fact, we have this little multiplication table for tensor prod
algebras:


tensor R C H
----------------------
R | R C H
|
C | C C+C C(2)
|
H | H C(2) R(4)


Anyone with an algebraic bone in their body should spend an afternoon
figuring out how this works! But I won't explain it now.

Instead, I'll just note that the complex numbers are very aggressive and
infectious - tensor anything with a C in it and you get more C's. That's
because they're a field in their own right - and that's why they don't
live in the Brauer group of the real numbers.

They do, however, live in the *super-Brauer* group of the real numbers,
which is Z/8 - the Clifford clock itself!

But before I explain that, I want to show you what the categories of
representations of the Clifford algebras look like:

0

real vector spaces

7 1
split real vector spaces complex vector spaces




6 real vector spaces quaternionic vector spaces 2




complex vector spaces split quaternionic vector spaces
5 3


quaternionic vector spaces

4


You can read this information off the 8-hour Clifford clock I showed you
before, at least if you know some stuff:

A real vector space is just something like R^n
A complex vector space is just something like C^n
A quaternionic vector space is just something like H^n

and a "split" vector space is a vector space that's been written as the direct
sum of two subspaces.

Take C_4, for example - the Clifford algebra generated by 4 anticommuting
square roots of -1. The Clifford clock tells us this is H + H. And if you
think about it, a representation of this is just a pair of representations of
H. So, it's two quaternionic vector spaces - or if you prefer, a "split"
quaternionic vector space.

Or take C_7. The Clifford clock says this is R + R... or at least Morita
equivalent to R + R: it's actually R(Cool + R(Cool, but that's just a beefed-up
version of R + R, with an equivalent category of representations. So, the
category of representations of C_7 is *equivalent* to the category of split
real vector spaces.

And so on. Note that when we loop all the way around the clock, our
Clifford algebra becomes 16 x 16 matrices of what it was before, but this
is Morita equivalent to what it was. So, we have a truly period-8 clock
of categories!

But here's the really cool part: there are also arrows going clockwise and
counterclockwise around this clock! Arrows between categories are called
"functors".

Each Clifford algebra is contained in the next one, since they're built
by throwing in more and more square roots of -1. So, if we have a
representation of C_n, it gives us a representation of C_{n-1}. Ditto
for maps between representations. So, we get a functor from the category
of representations of C_n to the category of representations of C_{n-1}.
This is called a "forgetful functor", since it "forgets" that we have
representations of C_n and just thinks of them as representations of C_{n-1}.

So, we have forgetful functors cycling around counterclockwise!

Even better, all these forgetful functors have "left adjoints" going
back the other way. I talked about left adjoints in "week77",
so I won't say much about them now. I'll just give an example.

Here's a forgetful functor:

forget complex structure
complex vector spaces ---------------------------> real vector spaces

which is one of the counterclockwise arrows on the Clifford clock.
This functor takes a complex vector space and forgets your ability to multiply
vectors by i, thus getting a real vector space. When you do this to C^n,
you get R^{2n}.

This functor has a left adjoint:

complexify
complex vector spaces <-------------------------- real vector spaces

where you take a real vector space and "complexify" it by tensoring it with
the complex numbers. When you do this to R^n, you get C^n.

So, we get a beautiful version of the Clifford clock with forgetful functors
cycling around counterclockwise and their left adjoints cycling around
clockwise! When I realized this, I drew a big picture of it in my math
notebook - I always carry around a notebook for precisely this sort of thing.
Unfortunately, it's a bit hard to draw this chart in ASCII, so I won't
include it here.

Instead, I'll draw something easier. For this, note the following mystical
fact. The Clifford clock is symmetrical under reflection around the
3-o'clock/7-o'clock axis:


0

real vector spaces

7 1

split real vector spaces complex vector spaces
\
\
\
\
6 real vector spaces \ quaternionic vector spaces 2
\
\
\
\
complex vector spaces split quaternionic vector spaces
5 3


quaternionic vector spaces

4

It seems bizarre at first that it's symmetrical along *this* axis instead
of the more obvious 0-o'clock/4-o'clock axis. But there's a good reason,
which I already mentioned: the Clifford algebra C_n is related to rotations in
n+1 dimensions.

I would be very happy if you had enough patience to listen to a full
explanation of this fact, along with everything else I want to say. But
I bet you don't... so I'll hasten on to the really cool stuff.

First of all, using this symmetry we can fold the Clifford clock in half...
and the forgetful functors on one side perfectly match their left adjoints
on the other side!

So, we can save space by drawing this "folded" Clifford clock:


split real vector spaces

| ^
forget splitting | | double
V |

real vector spaces

| ^
complexify | | forget complex structure
v |

complex vector spaces

| ^
quaternionify | | forget quaternionic structure
v |

quaternionic vector spaces

| ^
double | | forget splitting
v |

split quaternionic vector spaces


The forgetful functors march downwards on the right, and their
left adjoints march back up on the left!

The arrows going between 7 o'clock and 0 o'clock look a bit weird:


split real vector spaces

| ^
forget splitting | | double
V |

real vector spaces


Why is "forget splitting" on the left, where the left adjoints belong, when
it's obviously an example of a forgetful functor?

One answer is that this is just how it works. Another answer is that it
happens when we wrap all the way around the clock - it's like how going from
midnight to 1 am counts as going forwards in time even though the number is
getting smaller. A third answer is that the whole situation is so symmetrical
that the functors I've been calling "left adjoints" are also "right adjoints"
of their partners! So, we can change our mind about which one is
"forgetful", without getting in trouble.

But enough of that: I really want to explain how this stuff is related
to the super-Brauer group, and then tie it all in to the *topology* of Bott
periodicity. We'll see how far I get before giving up in exhaustion....

What's a super-Brauer group? It's just like a Brauer group, but where we
use superalgebras instead of algebras! A "superalgebra" is just physics
jargon for a Z/2-graded algebra - that is, an algebra A that's a direct
sum of an "even" or "bosonic" part A_0 and an "odd" or "fermionic" part A_1:

A = A_0 + A_1

such that multiplying a guy in A_i and a guy in A_j gives a guy in A_{i+j},
where we add the subscripts mod 2.

The tensor product of superalgebras is defined differently than for algebras.
If A and B are ordinary algebras, when we form their tensor product, we
decree that everybody in A commutes with everyone in B. For superalgebras
we decree that everybody in A "supercommutes" with everyone in B - meaning
that

ab = ba

if either a or b are even (bosonic) while

ab = -ba

if a and b are both odd (fermionic).

Apart from these modifications, the super-Brauer group works almost like the
Brauer group. We start with superalgebras over our favorite field - here
let's use the real numbers. We say two superalgebras are "Morita equivalent"
if they have equivalent categories of representations. We can multiply
these Morita equivalence classes by taking tensor products, and if we just
keep the invertible classes we get a group: the super-Brauer group.

As I've hinted already, the super-Brauer group of the real numbers is Z/8 -
just the Clifford algebra clock in disguise!

Here's why:

The Clifford algebras all become superalgebras if we decree that all the
square roots of -1 that we throw in are "odd" elements. And if we do this,
we get something great:

C_n tensor C_m = C_{n + m}

The point is that all the square roots of -1 we threw in to get C_n
*anticommute* with those we threw in to get C_m.

Taking Morita equivalence classes, this mean

[C_n] [C_m] = [C_{n+m}]

but we already know that

[C_{n+8}] = [C_n]

so we get the group Z/8. It's not obvious that this is *all* the super-Brauer
group, but it actually is - that's the hard part.

Now let's think about what we've got. We've got the super-Brauer group,
Z/8, which looks like an 8-hour clock. But before that, we had the categories
of representations of Clifford algebras, which formed an 8-hour clock with
functors cycling around in both directions.

In fact these are two sides of the same coin - or clock, actually. The
super-Brauer group consists of Morita equivalence classes of Clifford
algebras, where Morita equivalence means "having equivalent categories
of representations". But, our previous clock just shows their categories
of representations!

This suggests that the functors cycling around in both directions are secretly
an aspect of the super-Brauer group. And indeed they are! The functors going
clockwise are just "tensoring with C_1", since you can tensor a representation
of C_n with C_1 and get a representation of C_{n+1}. And the functors going
counterclockwise are "tensoring with C_{-1}"... or C_7 if you insist, since
C_{-1} doesn't strictly make sense, but 7 equals -1 mod 8, so it does the
same job.

Hmm, I think I'm tired out. I didn't even get to the topology yet! Maybe
that'll be good as a separate little story someday. If you can't wait,
just read this:

4) John Milnor, Morse Theory, Princeton U. Press, Princeton, New Jersey, 1963.

You'll see here that a representation of C_n is just the same as a vector
space with n different anticommuting ways to "rotate vector by 90 degrees",
and that this is the same as a real inner product space equipped with a map
from the n-sphere into its rotation group, with the property that the north
pole of the n-sphere gets mapped to the identity, and each great circle
through the north pole gives some action of the circle as rotations. Using
this, and stuff about Clifford algebras, and some Morse theory, Milnor gives a
beautiful proof that

Omega^8(SO(infinity)) ~ SO(infinity)

or in English: the 8-fold loop space of the infinite-dimensional rotation
group is homotopy equivalent to the infinite-dimensional rotation group!

The thing I really like, though, is that Milnor relates the forgetful functors
I was talking about to the process of "looping" the rotation group. That's
what these maps from spheres into the rotation group are all about... but I
want to really explain it all someday!

I learned about the super-Brauer group here:

5) V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction,
American Mathematical Society, Providence, Rhode Island, 2004.

though the material here on this topic is actually a summary of some
lectures by Deligne in another book I own:

6) P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan,
D.R. Morrison and E. Witten, Quantum Fields and Strings: A Course For
Mathematicians 2 vols., American Mathematical Society, Providence, 1999.
Notes also available at http://www.math.ias.edu/QFT/

Varadarajan's book doesn't go as far, but it's much easier to read, so I
recommend it as a way to get started on "super" stuff.

-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twf.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html
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robert bristow-johnson
science forum Guru Wannabe


Joined: 02 May 2005
Posts: 105

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: A potential criticism of the anthropic principle in cosmology Reply with quote

i hope this is acceptable to the moderators of s.p.r since it gets more into
the philosophy and less into tangible physics. not necessarily picking on
Alex, but he said a bunch that i wanted to comment on.

in article 1109958658.462932.130420@z14g2000cwz.googlegroups.com, Alex at
dralexgreen@yahoo.co.uk wrote on 03/06/2005 02:50:

Quote:
I share your disquiet about the anthropic principle but for different
reasons.

i have no problem the *weak* anthropic principle, which does appears to me
as a tautology. but just because it *is* a tautology doesn't bother me.
it's very well likely to be also true (a truism), even if it doesn't really
say much.

Quote:
Throughout history people have thought things were so complicated God
or gods must have made them and have only been shifted from this belief
as knowledge progressed. Cicero, a sensible academic, pointed out this
foible of humanity 2000 years ago and nothing has changed except there
are fewer things that are unexplained (cf: 'On the Nature of the
Gods'). Gods are indeed a possible generic explanation for complicated
events but as yet other explanations have nearly always been found.

to *some* extent. science explains a lot but there are philosophical
observations (you know, the standard stuff regarding consciousness,
aesthetics, values, good & evil) that are not really in the purview of
science. science really cannot explain those realities, because it is about
reality that we can actually physically observe and measure and, if
repeatable, develop theories that may predict such outcomes and thus can be
tested.

even stuff that *is* in the purview of science might never be explained such
as why would the big bang ever bother to bang (or why the multiverse should
bother to exist, if you believe in that) or even the fine-tuning of the
universe. and it seems to me that as time and science progress, that many
old questions are answered (without appealing to the supernatural) but even
more new questions present themselves. we'll always have a lot of
unexplained phenomena, even if a lot of it has been explained.

it is to this apparent unexpected order (that structures such as stars exist
in such a way so that reasonably heavy elements can exist) and fortuitous
selection of physical parameters (what would happen if alpha was 1/13
instead of 1/137?) that prompts the "intelligent design" postulation (by
theists, almost exculsively) and the resort to the anthropic principle
(which one is often not explicit) of others to explain it away.

even arguments for the evolution of consciousness in higher animals in order
for them to encompass or encapsulate all of these previously unconscious
brain functions, in order to provide some "executive overview" that allows
that animal to put stimulus together and compete more successfully in the
biological marketplace, that explanation, while kinda satisfying, does not
explain what, physically, our deepest thoughts and feelings and aspirations
are. science cannot explain why herding undesirable people into gas
chambers or crushing babies' skulls with rifle butts is "immoral" or even
what morality is. whether or not any gods exist, i would bet that these
subjects of reality will remain forever outside of the reach of science,
even though i won't be around long enough to pay off the bet should i lose.

Quote:
In modern times the laws of chance are a similar generic 'explanation'
and we can suggest that if there are an infinity of possible processes
and phenomena available then any outcome is possible. So, to me the
anthropic principle is like invoking Intelligent Design, it is a
generic explanation that does not really explain anything.

a tautology does not really explain any new knowledge, but if it's really
*is* a tautology (or a "truism"), it might help people remember what is
salient. and it's true, even if it doesn't say much. it's like saying
5 = 5 or x*y = 5 and (x+y)^2 - x^2 - y^2 - 10 = 0 which is true but doesn't
tell you much about x or y.

i dunno if this is a reference to the multiverse theory of which our
universe with all its special order and parameters is just one of the
unlikely possible outcomes, that explanation cannot be falsified or
verified. so it remains a (speculative) "theory" that can rightly be
criticized in the way that the "theory" of evolution is wrongly attacked.
even though none of us were around to see the evolution of species, there is
a lot of evidentiary support to Darwinism and there is nothing, other than
the speculation of theorists, that can support or refute the existence of
other universes, some or most that might not be as lucky as ours yielding no
conscious life.

the so-called "Strong anthropic principle" seems to invoke Intelligent
Design, and is not, in my opinion, a tautology. but because of what it
really says or adds to the discussion, it is a bona-fide explanation
(whether it is true or not) that i do not believe should ever be taught in a
science class. it isn't science, but often a euphemism for Creationism (at
least for Theism), which (speaking as a theist and churchgoer) is a blatant
attempt to inject specific faiths and NON-scientific process into the
science classroom. theology is not science nor vise-versa even if one
(science) can sometimes inform the other (theology).

..

Quote:
Our physiology is not as far from the physiology of a frog as we would
like to believe.

i remember pithing and dissecting frogs in high school biology and being
told there *was* a lot of common physiology to humans. that, and also that
they were relatively cheap, was the reason we crucified them poor amphibians
to the board. :-)

Quote:
Which brings us back to what we think is special about ourselves: we
possess conscious observation. However until we know the nature of
conscious observation we cannot use this 'specialness' in scientific
speculation because it may not be special at all.

well, it seems that cattle have some kinda consciousness and yet we make
hamburger out them. (and there are human cultures that object to that.)

Quote:
It may turn out that
roundworms have conscious observation. It is even possible that
panpsychism applies. The truth is that we do not know.

i think i'm safe in believing that neither mold nor trees nor dead human
bodies nor granite has consciousness. even though we do not know (and
likely will never know) precisely what consciousness is, in the physical
domain, we're pretty confident that there is some association with the
firing of neurons.

Quote:
Until we do know we can be certain that any conscious entity, no matter
what form it might take, will consider itself to be unique and special
and invoke the tautological argument that the universe was designed to
allow it to exist.

i don't think it's a tautological argument to say that the universe was
designed to allow conscious beings to exist (the strong anthropic
principle). that actually says something, and enough that an atheist might
have to disagree.

the tautology (in the _weak_ anthropic principle) is that, of course we can
only be around to observe and wonder and awe at this amazingly luckily
fine-tuned universe of which only such would allow beings like us to evolve
and exist to observe and wonder and awe at. it does not explain *why* there
is this order (other than the multiverse theory which is so speculative that
it leaves even more to explain) but reminds us that if those "special"
circumstances weren't there, we wouldn't be around (at least in this form)
to observe and wonder and comment about them. i think that this particular
tautology is something that both theists (that accept the scientific method)
and non-theists can accept, since, as a tautology, it kinda has to be true,
even if it doesn't say much.

--

r b-j rbj@audioimagination.com

"Imagination is more important than knowledge."
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Arnold Neumaier
science forum Guru


Joined: 24 Mar 2005
Posts: 379

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

Eugene Stefanovich wrote:
Quote:
Arnold Neumaier wrote:

Eugene Stefanovich wrote:

I am reading some papers trying to grasp the idea of the "closed path
formalism". I cannot understand the following thing: In quantum
theory the time evolution is described by the standard time
evolution operator, i.e., the exponent of the Hamiltonian.
Why someone needs to invent the new "closed path formalism"?

Because relativistic field theory is defined in terms of
actions which are covariant, and picking a Hamiltonian destroys
the manifest symmetry and makes everything look messy.

This is not the case if you
are interested in time evolution. Then you need an explicit expression
for the Hamiltonian (and for 9 other generators of the Poincare group,

There is such an expression for all generators of the Poincare group
in every P-invariant field theory, no matter which form of dynamics
is used.


Quote:
The manifest covariance is not a useful physical principle.
That's the whole point of my book.

But physicists apart from you find it very useful.

And they use the CTP formalism successfully for many problems for which
you don't have an answer. So you should not dismiss their approach.


Arnold Neumaier
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Igor Khavkine
science forum Guru


Joined: 01 May 2005
Posts: 607

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

On Thu, 03 Mar 2005 22:25:43 +0000, Sci~Girl wrote:
Quote:
Igor Khavkine wrote...

Does this answer your question?

Yes. It also gives me some insight into how ridiculously easily answered
all my questions are... they cause me so much confusion and yet the
answers are so obvious.

All questions are non-trivial until answered and obvious afterward.

Quote:
I have another one though, forgive me if it's also that elementary... I
read that the leptons correspond one for one with the quarks. Does this
mean each has the same mass and charge, and spin as a quark, and if so,
why is it that quark masses were revealed much more easily than lepton
masses?

Eh, that is not true. Elementary particles are grouped into generations.
Each generation has two leptons and two quarks. Perhaps you can call this
grouping "correspondence". For example, the first generation has the
electron, the electron neutrino, the up quark and the down quark. Each of
these particles has different charges for the electric, weak, and strong
forces. Each fundamental fermion that we know of (lepton or quark) carries
spin 1/2.

Lepton masses are quite well known (for instance, the electron mass has
been known for a long time) while quark masses are harder to determine
because they can only be found bound inside baryons (composite particles
such as protons, neutrons, and others). But quark masses have also been
determined.

Hope you can extract answers to your question from the above.

Igor
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Igor Khavkine
science forum Guru


Joined: 01 May 2005
Posts: 607

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

On Wed, 02 Mar 2005 07:04:50 +0000, Eugene Stefanovich wrote:

Quote:
I wasn't clear enough in my previous post, so I better clarify my position
here. I do not question the ability of the closed path formalism to
describe the time evolution at non-zero temperatures if the system is
described by a well-defined Hamiltonian. I am not interested in non-zero
temperatures. I am interested in the time evolution within standard QED.
The problem is that the Hamiltonian of QED (expressed through bare
particle operators) is infinite, and I do not see how it can be used to
calculate the time evolution unless we either redefine the Hamiltonian or
redefine the particles. I had a feeling (please correct me if I am wrong)
that you suggest the closed path formalism as a way to do the time
evolution at T=0 in QED with this divergent Hamiltonian. That's the part I
do not understand.

At the risk of reignighting this old discussion, I must once again point
out that the QED Hamiltonian is not infinite. The infinities you refer to
are simply an artifact of the standard choice of (formal) basis in the QED
Hilbert space. Your approach amounts to a change of basis where the
(perturbative) finiteness of the Hamiltonian is more obvious. The results
you obtain from your calculations can only be at odds with the standard
formulation of QED only in your interpretation (which is not physically
significant).

BTW, finite temperature and finite time calculations can be analytically
continued into each other.

Igor
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Eugene Stefanovich
science forum Guru


Joined: 24 Mar 2005
Posts: 519

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

Franz Heymann wrote:
Quote:
"Eugene Stefanovich" <eugenev@synopsys.com> wrote in message
news:42164384.3090301@synopsys.com...

[snip]


There is a more consistent way to look at interactions: QFT can be
reformulated in terms of real particles and instantaneous potentials
acting between them.
All experimental predictions (e.g., the S-matrix) remain the same,
and quite a lot of "invisible" stuff gets removed from the theory:
No retardation, no virtual particles, no fields.


I'll go for the virtual particles rather than some mysterious action
at a distance.


There is a way to distinguish these two approaches in experiment.
Take two charged macroscopic particles (e.g., two specks of dust)
in vacuum. Arrange a slow collision of these particles and measure
their trajectories with a good time resolution. We are interested,
in particular, in the dependence of particle momenta on time.
At the same time measure momenta of all real photons emitted by this
couple of charges (there could be "soft" photons whose detection
is tricky). If there are virtual particles, we should see some imbalance
of the total momentum (specks of dust + photons): a part of the total
momentum belongs to invisible virtual particles which carry the
interaction between the two charges. If there are no virtual particles
(interaction propagates instantaneously) we should see that the
total momentum of real particles is conserved at all times, because
there are no "virtual" degrees of freedom which can carry the extra
momentum.

I am not sure if current experimental resolution is good enough
to figure out which of the two approaches is better.

Eugene Stefanovich.
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Eugene Stefanovich
science forum Guru


Joined: 24 Mar 2005
Posts: 519

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

Arnold Neumaier wrote:
Quote:
Eugene Stefanovich wrote:



I had a feeling (please correct me if I
am wrong) that you suggest the closed path formalism as a way
to do the time evolution at T=0 in QED with this divergent
Hamiltonian.


One can take the limit T--> 0 and then get vacuum expectations.
Of course, CTP must also be renormalized to get well-defined results.

I find it rather odd that one first needs to formulate a
theory at non-zero temperature and then take a limit T--> 0
in order to describe the time evolution for simple systems
like collision of two electrons, where temperature plays no
role at all.

I see a better way:
1) derive finite "renormalized" Hamiltonian either via
Glazek-Wilson "similarity renormalization" or via my
"dressing+similarity" approach.
2) Obtain a well-defined time evolution operator by
taking an exponent of the Hamiltonian.
Quote:



I do see now how Glazek-Wilson "similarity renormalization"
leads to a finite Hamiltonian which, in principle, can be
plugged into the time evolution operator.


Their technique is applied to do real calculations, which are
obviously finite. Of course approximate, but so is yours,
since you can calculate only at some fixed order.


The problem with this
approach is that there is still a non-trivial dressing
(the difference between bare and dressed particles) and
(finite) mass renormalization, i.e. the mass of dressed particles is
different from the mass of bare particles whose
operators are present in the Hamiltonian.


The bare masses are functions of the dressed masses and the cutoff,
chosen such that the limit for cutoff to infinity exists.
There is nothing wrong with that.

There IS a problem with QFT approaches requiring renormalization:
they predict clouds of virtual particles. Nobody have ever seen
virtual or bare particles in experiment. On the other hand,
there are no bare and virtual particles in RQD: just real physical
particles and instantaneous forces between them. Both approaches
predict the same S-matrix, but their predictions for time-dependent
processes are different. Unfortunately, currently there are no
time-resolved experiments in high-energy physics. So we should wait
a while until the differences between the two theories can be
tested by experiment.

Eugene Stefanovich.
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Eugene Stefanovich
science forum Guru


Joined: 24 Mar 2005
Posts: 519

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

Arnold Neumaier wrote:
Quote:
Eugene Stefanovich wrote:


I am reading some papers trying to grasp the idea of the "closed path
formalism". I cannot understand the following thing: In quantum
theory the time evolution is described by the standard time
evolution operator, i.e., the exponent of the Hamiltonian.
Why someone needs to invent the new "closed path formalism"?


Because relativistic field theory is defined in terms of
actions which are covariant, and picking a Hamiltonian destroys
the manifest symmetry and makes everything look messy.


You can stay within the manifestly covariant formalism if you
are interested only in properties related to the S-matrix.
Then you can start from action and Lagrangian, directly derive
the Feynman rules from there, and calculate the S-matrix which
is itself manifestly covariant (it has simple Lorentz transformations
wrt boosts). There is no need for the Hamiltonian.

This is not the case if you
are interested in time evolution. Then you need an explicit expression
for the Hamiltonian (and for 9 other generators of the Poincare group,
including 3 boost operators which contain interaction terms).
This formulation loses the manifest covariance, but it still
has perfect relativistic invariance, because all 10 generators
satisfy the Poincare commutation relations. You may call this
formalism messy, but I call it correct. I do not know any
consistent formalism which
1) describes the time evolution 2) keeps manifest covariance
(= simple linear Lorentz transformation of observables wrt
boosts of observers). The presence of interaction terms in the
boost operator does not permit condition 2). This is the Currie-
Jordan-Sudarshan theorem.

The manifest covariance is not a useful physical principle.
That's the whole point of my book.

Eugene Stefanovich.
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FrediFizzx
science forum Guru


Joined: 01 May 2005
Posts: 774

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

"mathman" <mathnucl@optonline.net> wrote in message
news:mathman.1labsu@physicsforums.com...
| There are at least 2 classes of virtual particles. First there are
| particles associated with the vacuum of space - popping in and out of
| existence all of the time. A major observation of their presence is
by
| the Casimir effect. Look it up in Google and get a description of
what
| it is all about. Second there are particles associated with particle
| interactions. A simple example is the repulsive force between
| electrons, described by photons going between them.

Aren't your "2 classes" just virtual fermions and virtual bosons?

FrediFizzx
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Ian Parker
science forum Guru Wannabe


Joined: 15 Jul 2005
Posts: 100

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: A potential criticism of the anthropic principle in cosmology Reply with quote

It is always possible to meet any cooincidence by postulating a large
number of Universes. In fact the present Universe is infinite. It is
finite in time of course but as the apparent edge is approached the
Cosmological principle of being apparently in the center is obeyed.
This is in fact the mathematical definition of infinity. No matter how
far we go we can always go further. This has the consequence that any
inconsistency in the statistics of something like evolution can be met
by simply postuating an infinite Universe. In an infinite Universe any
number of heads will always come in sequence.

Thus an infinite number of infinite universes will imply the Anthropic
principle.

To me this is deeply unsatisfacory and perhaps the definition of bad
science. Scientific explainations should in my view avoid infinite
sequences of heads. I also happen to believe that human intelligence
could not have evolved in the time it did without a sequence of heads.
Why do we struggle so much with AI?

josephus <dogbird@earthlink.net> wrote in message news:<jT6Pd.6008$mG6.39@newsread1.news.pas.earthlink.net>...
Quote:
Bill wrote:

roninfromde@yahoo.com wrote:



I think your ideas are really just rationalizations. To go with a
theory b/c the other side has 1 more falsifiable claim seems less than
scientific to me.


It's called Occam's razor, actually. Useful but not necessary for
scientific work.

I guess there could be some remotely interesting

esoteric discussion there, but I'd rather debate the viability of a
claim, not how many adjuncts of it are probable or whatever.


Actually I like the idea of multiple universes a lot, from the
standpoint of having room for lots of fantastic places. It just seems,
unfortunately, to add unnecessary complexity to me, at the expense of
looking for an underlying physics that might allow us, for example, to
change a constant like g to our own advantage Smile.

--Bill

I dont see anybody talking about the antrhopomorphic princilple. My
limited undertstannding is that if we assume that the 'Laws' of the
univers are not fixed. There maybe other universes that have different
basics laws. There have been some theoretical models that say a small
change would absolutely elmimate any chance of life. Examples. if the
weak force were to change by 10%. anything change coupling change.

But in a problem closer to home. We live in a buble of tranquitllty.
we can see the universe is a violet and deadly place. A supernove
within 50 ly would incerate the earth. The fact that we do not have
such violent events is that our environment is tailored to support us.
that is the anthropomorhic principle. It says nothing about why but
trys to understand what.
I really have only a limited understanding.
josephus
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Sci~Girl
science forum beginner


Joined: 08 May 2005
Posts: 15

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

Igor Khavkine wrote...
Quote:
Masslessness, in the standard meaning of the word, is equivalent to
motion
with the speed of light (and only the speed of light). Photons travel
only
at the speed of light, thus they are massless. All fermions that we
know
(electrons, protons, neutrinos) seem to have mass (i.e. cannot travel
at
the speed of light).

Does this answer your question?

Yes. It also gives me some insight into how ridiculously easily
answered all my questions are... they cause me so much confusion and
yet the answers are so obvious.

I have another one though, forgive me if it's also that elementary...
I read that the leptons correspond one for one with the quarks. Does
this mean each has the same mass and charge, and spin as a quark, and
if so, why is it that quark masses were revealed much more easily than
lepton masses?
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ohwilleke
science forum beginner


Joined: 08 May 2005
Posts: 2

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: non-GR theories of gravity Reply with quote

In support of the proposition that the Rank 2 tensors are not identical
see this article:
http://www.arxiv.org/abs/gr-qc/0409089

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This post submitted through the LaTeX-enabled physicsforums.com
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Peter Tobias
science forum beginner


Joined: 08 May 2005
Posts: 3

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: Basic Statistical Mechanics Questions Reply with quote

lost.and.lonely.physicist@gmail.com:
Quote:
1) Can thermodynamics be completely derived from statistical
mechanics?
For instance, the first law of thermodynamics states
dE = T dS - P dV + mu dN
where E is energy, T temperature with the Boltzmann's constant = 1, S
entropy, P pressure, V volume, mu the chemical potential, and N the
number of particles.

The first law in this form is just another form to postulate energy
conservation. Statitical mechanics doesn't derive energy conservation
either, but postulates it in a another form.

Quote:
a) How does one know there aren't any more terms in this "1-form"
expansion? How do we know a system can be described by T, P, V, and N
only; no other possible variables other than re-expressing them in
terms of S, E, mu, etc.?

One doesn't know in advance but assumes. If this form doesn't fit to
an experiment, against our assumption, we would search for other terms.
Only if this search is fruitless, energy conservation would be
questioned.

Quote:
Thanks for the help and I hope this group will tolerate my elementary
questions - I may have more to add later.

Elementary questions started science and can lead to good discussions.
Regards,

Peter
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MobyDikc
science forum Guru Wannabe


Joined: 29 Apr 2005
Posts: 170

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: Predictions from Mathematical models Reply with quote

Mark Palenik wrote:
Quote:
"Mike Helland" <mobydikc@gmail.com> wrote in message
news:1107041709.049901.21570@z14g2000cwz.googlegroups.com...
Premises:

snip
4. There is nothing about physics, or the scientific method, that
says
this is the one and only way of deriving predictions from
mathematical
models.
snip

Here is the idea that I came up with:

Instead of taking a value of of the system, we need to investigate
the
system to find out what observers in the system know about the
system.

What do I mean by that? Say you have some modeled electrons and
protons
and you arrange them into a mercury thermometer.


___oo__oo___ooo
___oo__oo
___oo__oo___oo
___oo__oo
___oo++oo___o
__oo++++oo
__oo++++oo
___oooooo

When the motion of the electrons increases, the space they take up
increases, and the level of the mercury rises in the thermometer.

The values of the mathematical model only represent properties of
particles, and there is no temperature for an individual particle.

That's right. Although, mathematically, you could create a model
where a
single particle had temperature, like if you actually defined
temperature as
the average kinetic energy of the particles present. It just turns
out that
the concept itself (of a single particle with temperature) is
meaningless,
and that 1/(d_sigma/d_U) is very meaningful.


But if we were to arrange these values in some graphical
representation
like the one above, we could make the analysis the the level of the
thermometer is at the single "o" mark.

So, in the units of temperature that this thermometer is measuring,
the
temperature in the model is 1 degree.

We've derived the prediction "1 degree" from our model, but the
value
"1" does not exist in the model itself. The only values in the
model
represent the properties of the particles.

Your model is that T~V, for a fixed number of particles. Of course,
for an
ideal gas, we know that PV = NKT, so this prediction doesn't sound
too far
off the mark.

I don't see what's so new here. A physicist would take an idea like
the one
you've presented, then write it down mathematically, using
experiments to
figure out exactly what temperatures the little "o"s correspond to
and what
the exact relationship is between all of the variables.

So far, we don't really have anything that can be used to predict
real world
phenomena, other than T~V. You, perhaps unwittingly, have given a
mathematical relationship. However, we can't determine what that
exact
relationship is without experiments.


This method of deriving predictions is new. No scientific theory
works
this way.


I'm sorry, but I don't really see what this new thing you're trying
to get
at is. If you're trying to say that all of physics can be derived
logically
rather than empirically, that's provably false. It's also
empirically
false, since people who have tried to do this, like Aristotle, have
failed.
That's not to say that there aren't things that can be determined
logically,
but exact values and fundimental principles need to be found
experimentally.


How this applies to temperature doesn't seem very radical. I agree.

But what if this is applied to how we measured velocity?

Right now we make measurements of time and distance, and plug them into
a mathematical model (d/t, pretty easy) and that is velocity.

I assume you agree with my conclusion, that nothing about science
(except for tradition) says that this is the only way to use
mathematical models (input what is known, solve for what is not known).
Correct?

In the new suggested approach, you do not measure time or distance and
input them. Instead, you have a mathematical model with values as
initial conditions and rules that change the values.

Those values, it is postulated, describe the properties of particles.

We must configure those values, which effectively 'arranges' the
particles they represent, into a structure that is capable of observing
the mathematical model in which the structure exists.

This observer will then perform measurements of space and time, and
then will say that "in 5 ticks of my watch, the object moved 6
distances of my rules", which is effectively velocity.

In this approach to science, the values, the inputs, are not comparable
to the outcome of measurement. Instead, only what is measured from
within the mathematical model itself, by an observer who exists and
functions purely within the mathematical model, is comparable to the
outcome of our measurements.

It is a very simple idea, but it is very very different from anything
physics has attempted so far. So the stronger your inution towards
current physics is, the harder this will be to comprehend.

But if you think about it, this is exactly the type of innovation to
mathematical models that quantum and realtivistic phenomena cry out for.
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Arnold Neumaier
science forum Guru


Joined: 24 Mar 2005
Posts: 379

PostPosted: Sun May 08, 2005 6:46 pm    Post subject: Re: How real are the "Virtual" partticles? Reply with quote

Eugene Stefanovich wrote:

Quote:
There IS a problem with QFT approaches requiring renormalization:
they predict clouds of virtual particles.

No. At worst they use virtual particles in intermediate calculations.

What they predict are instead cross sections, quantum Boltzmann
equations, and the like, which are in agreement with experiment.

I wonder how _you_ predict a relativistic quantum Boltzmann equation
for an plasma in an electromagnetic field, which is surely part of QED.


Arnold Neumaier
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