Particles and Group Representations

[willo_thewisp@hotmail.com]

I wonder if someone can suggest the perfect book for me to read.

My goal is to have a clear conceptual understanding of what it means
to say that a particle "is" an irreducible group representation.

Here is what I already do and do not know:

1) I have a rudimentary understanding of quantum mechanics, at the
level, say, of the first few chapters of the Cohen-Tannoudji/Diu/Laloe
textbook. That is, I understand the formalism of bras and kets, the
basic postulates, and the Schrodinger equation.

2) I at one time understood Noether's theorem on symmetry and
conservation laws and I assume I could learn it again, though I've
misplaced much of my past understanding.

3) I understand the basics of group theory, representation theory
and character theory. I have a pretty thorough understanding of the
theory for finite groups. My knowledge of Lie Groups is pretty
limited---I am not sure what the Borel-Weil theorem says, etc.---but
I feel confident I can learn this.

4) I know about manifolds, metrics, connections, vector bundles and
principal bundles. Here I speak the language of mathematics but I
can presumably learn to translate from the language of physics.

5) I have a (fairly recently acquired) reasonably good understanding
of Maxwell's equations, in case this is relevant.

6) I understand how various Lie groups act on the state spaces
of quantum mechanical systems, and that I can think of these actions
as representations of the Lie groups (either on vector spaces or
quotient of vector spaces). I understand that I can write these
representations as direct sums of irreducible representations. I have
the vague sense that this is somehow related to what I really want
to understand, which is:

A) How does an irreducible representation yield a particle? And
B) how do the properties of that representation predict properties
of the particle, e.g. its mass?


I am more interested in understanding all this at a conceptual level
than in being able to solve problems and do calculations. I will
probably be more comfortable with a text written for mathematicians
than a text written for physicists, but I can deal with either.

[Arnold Neumaier]

willo_thewisp@hotmail.com schrieb:
> I wonder if someone can suggest the perfect book for me to read.
>
> My goal is to have a clear conceptual understanding of what it means
> to say that a particle "is" an irreducible group representation.
>
> 3) I understand the basics of group theory, representation theory
> and character theory. I have a pretty thorough understanding of the
> theory for finite groups. My knowledge of Lie Groups is pretty
> limited
>
> A) How does an irreducible representation yield a particle? And

Whatever deserves the name ''particle'' must move like a single,
indivisible object. The Poincare group must act on the description of
this single object; so the state space of the object carries a
unitary representation of the Poincare group. This splits into a direct
sum or direct integral of irreducible reps. But splitting means
divisibility; so in the indivisible case, we have an irreducible
representation.

On the other hand, not all irreducible unitary reps of the Poincare
group qualify. Associated with the rep must be a consistent and causal
free field theory. As explained in Volume 1 of Weinberg's book on
quantum field theory, this restricts the rep further to those with
positive mass, or massless reps with quantized helicity.

Much more details can be found in Section S2c (Representations of
the Poincare group, spin and gauge invariance) of my theoretical
physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt
In the other sections with labels S2* there is further related material.

> B) how do the properties of that representation predict properties
> of the particle, e.g. its mass?

The physically relevant irreducible reps come in families parameterized
by mass and spin(if mass>0) or helicity (if mass=0), which have the
familiar physical meaning.

> I am more interested in understanding all this at a conceptual level
> than in being able to solve problems and do calculations. I will
> probably be more comfortable with a text written for mathematicians
> than a text written for physicists, but I can deal with either.

Then my book
Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras,
Cambridge University Press, to appear (2009?).
http://www.mat.univie.ac.at/~neum/papers/physpapers.html#QML
arXiv:0810.1019
could be the right starting point, since the relevant Lie theory
is explained in detail and related to physics. Section 1.6 and
the final chapter contain much of what you want to know.

Arnold Neumaier

[Stephen Blake]

willo_thew...@hotmail.com wrote:

> A) How does an irreducible representation yield a particle? And
> B) how do the properties of that representation predict properties
> of the particle, e.g. its mass?

Hi, I'm very interested in this topic, so here is my understanding.

A) The states of a quantum system are in a big Hilbert space H. The
states seen by different observers are related by a unitary
representation U(g) of (say) the Poincare group. If Alice sees a state
|psi> then Bob sees this as U(g)|psi>. In general |psi> is sent to U
(g)|psi> with no obvious relation between the two states. However, the
big Hilbert space decomposes into a direct sum of smaller irreducible
subspaces as H=sum Lau where "a" labels the inequivalent subspaces and
"u" labels the copies. So, the states which span the subspace Lau are |
a,j,u> for j=1,2,...,na. Now, if Alice prepares a system in a state |
a,i,u> then Bob sees this system in a state U(g)|a,i,u> and Bob's
state remains in the small subspace Lau because U(g)|a,i,u> is just a
linear combination of the |a,j,u> for different j. So, if a state
belongs to an irreducible subspace Lau it is the most elementary sort
of system that can be prepared. Such a system is called a "particle"
because the states in the subspace Lau constitute the smallest set of
states which Alice and Bob and all other inertial observers (i.e.
observers whose perceptions are related by the Poincare group) can
agree to call different views of the same thing.

B) What we call "properties" of a particle or an elementary system
form the label "a" of the irreducible subspace Lau. So, if one finds
the irreducible subspaces of the Poincare group the states turn out
to be |M,s;p,m>. Here, M=mass and s=spin label the inequivalent
irreducible subspaces (i.e. M and s are the properties), whilst the 4-
momentum p and the z-component of spin m=s,s-1,...,-s label the states
within the irreducible subspace. I'm not sure if there are any copies
of these irreducible subspaces, I don't think so. The 4-momentum
labels p are the coords of points on an orbit p^2=M^2 and this is what
gives the clue that the label M is what we call mass.

Yours sincerely
Stephen
http://www.stebla.pwp.blueyonder.co.uk

[Stephen Blake]

willo_thew...@hotmail.com wrote:

> I wonder if someone can suggest the perfect book for me to read.

A.O.Barut and R. Raczka, Theory of Group Representations and
Applications, World Scientific, 2nd revised edition, 1986, 2000.

[Stephen Blake]

willo_thew...@hotmail.com wrote:

> I wonder if someone can suggest the perfect book for me to read.

A.O.Barut and R. Raczka, Theory of Group Representations and
Applications, World Scientific, 2nd revised edition, 1986, 2000.

[willo_thewisp@hotmail.com]

On Jan 14, 6:00*pm, Stephen Blake <ste...@blueyonder.co.uk> wrote:
> willo_thew...@hotmail.com wrote:
> > A) How does an irreducible representation yield a particle? *And
> > B) how do the properties of that representation predict properties
> > of the particle, e.g. its mass?
>
> Hi, I'm very interested in this *topic, so here is my understanding.
>
> A) The states of a quantum system are in a big Hilbert space H. The
> states seen by different observers are related by a unitary
> representation U(g) of (say) the Poincaregroup. If Alice sees a state
> |psi> then Bob sees this as U(g)|psi>. *In general |psi> is sent to U
> (g)|psi> with no obvious relation between the two states. However, the
> big Hilbert space decomposes into a direct sum of smaller irreducible
> subspaces as H=sum Lau where "a" labels the inequivalent subspaces and
> "u" labels the copies. So, the states which span the subspace Lau are |
> a,j,u> for j=1,2,...,na. Now, if Alice prepares a system in a state |
> a,i,u> then Bob sees this system in a state U(g)|a,i,u> and Bob's
> state remains in the small subspace Lau because U(g)|a,i,u> is just a
> linear combination of the |a,j,u> for different j. So, if a state
> belongs to an irreducible subspace Lau it is the most elementary sort
> of system that can be prepared. Such a system is called a "particle"
> because the states in the subspace Lau constitute the smallest set of
> states which Alice and Bob and all other inertial observers (i.e.
> observers whose perceptions are related by the Poincaregroup) can
> agree to call different views of the same thing.
>
>
Thanks. This was extremely helpful, in the "Oh! Now that you've
pointed this out I can't understand why it wasn't obvious!" sort
of way. I had somehow quite missed the point that irreducible
means "smallest subsystem that all inertial observers agree to
call different views of the same thing", and that this is what
a particle should be. This helps a lot.
>
Maybe you can help with one more thing: I understand that the
irreducible represenations are parameterized by invariants that get
labeled things like "mass" and "spin". But...what is the rule
that tells me which of these purely *mathematical* invariants
get matched up with which *physical* properties? Why is the
invariant identified with mass not identified instead with, say,
charge?
>
I will buy the book you recommended in your followup post, and
read it along with the Neumaier book recommended in the preceding
post. Thanks very much for this.

[Rock Brentwood]

On Jan 14, 5:00*pm, Stephen Blake <ste...@blueyonder.co.uk> wrote:
> Hi, I'm very interested in this *topic, so here is my understanding.
>
> A) The states of a quantum system are in a big Hilbert space H.

This statement is wrong -- not because what it says is wrong, but
because of what it does not say (and what it implies by virtue of not
saying it).

It's not "the states of a *quantum* system are in...", but "the states
of ALL systems are in ...". The distinction "quantum" is a red
herring.

There is nothing specifically classical or quantum in the use of
Hilbert spaces. Rather, they are an element of foundational physics
that cuts across the boundary between theories.

This is a small part of a much larger serious problem that pervades
the community from the elementary textbook level all the way up to the
level of the research literature. The problem doesn't have a name, so
I'll give it one and try to explain what I mean by it: "the Confusion
of Historical Change in Notation for Historical Change in Theory".

When Classical physics was formulated, in the time of Newton and
succeeding centuries, forms of mathematics were used that were widely
different than those that arose in the 19th and 20th centuries when
newer theories and paradigms also arose.

Many of these notations were motivated by the newer paradigms,
themselves. Consequently, they arose together. This, in turn, led to
widespread confusion whereby the notation, itself, was conflated with
the theory and paradigm.

The newer notation, however, is simply a consequence of the fact that
mathematics, itself, evolved by centuries in the span of time between
the emergence of the newer paradigms. In most (or all) cases, the
association was too "red herring", too incidential to be warranted,
but simply amounted to a case of:
(1) new math being applied to whatever new stuff came around to apply
it to anew
(2) while yet it was actually applicable across the board and
therefore had nothing specific to do with the "new stuff" at all.

The most important examples are: differential geometry <-> general
relativity; spacetime <-> special relativity, quantum theory <->
Hilbert spaces (along with the whole baggage of other related
concepts, such as Schroedinger picture, Heisenberg picture

So, let's bust open these wrong associations, one-by-one and debunk
the various myths about them that pervade the literature even now:

All physics systems have Hilbert space representations. This is
because Hilbert spaces are the linear spaces that house the
representations of C^*-algebras. These, in turn, embody the algebras
of observables for classical systems (commutative C*-algebras), pure
quantum systems (non-commutative C*-algebras with trivial centers) and
everything in between (non-commutative C*-algebras with non-trivial
centers).

Gravitational dynamics, Relativistic or non-relativistic, is
equivalently and naturally formulated in terms of the curved geometry
of space-time. Newton, Einstein or otherwise; it doesn't matter. This
is because of the "Equivalence Principle". In turn, the "Equivalence
Principle" has nothing per se to do with either General Relativity or
Lorentzian space-times. It's a general principle that cuts across the
board and states that free fall is locally equivalent to inertial
motion. The Galilean version of the Equivalence Principle was known
since the time of Galilei (and, in fact, well before then; at least as
far back as the 11th century).

The geometry underlying physics is that for space-time, not space.
This has nothing per se to do with the theories of Special and General
Relativity, nor the Minkowski geometry they are associated with.
Instead, where it comes from is the fact that the underlying
symmetries of space-time no longer segregate into purely spatial
symmetries and purely temporal symmetries.

Whereas you have the 6 degrees of purely spatial symmetry (linear and
angular translations) and the 1 degree of purely temporal symmetry
(time translation), you also have 3 more. They are the mixed
symmetries -- the Boosts. In turn, this addition arises from the
Principle of Relativity, which had also been known since the time of
Galileo (and well before then, amongst sailors and those who live in a
maritime world).

It is the principle of Relativity, a' la Galileo, that makes necessary
the replacement of purely spatial geometry by a deeper layer of
geometry in which the undefined element "point" has to be replaced by
"point-instant" or (in modern language) "event". The reason for this
is that once you make motion relative, this also makes genidentity
relative. But genidenity is the very prerequisite to the concept of
"point" itself.

This is, ultimately, why Newton balked at the idea of relativity. The
Principia is founded on *spatial* geometry. With Galilean relativity,
you (1) lose absolute rest (which is a fancy name for "genidentity"),
(2) then you lose genidentity, (3) then you lose the cohesiveness of
the very concept of "point". itself; (4) and thus spatial geometry is
incohesive; consequently (5) the foundation is cut from underneath the
Principia, itself.

(Genidentity is the name given to the relation that equates a point at
one time as being the "same point" as one at a later time. It is
equivalent to the concept of "rest" or being "stationary" or "being
at".)

The Galilei term ("-vt") in the transformation "x -> x - vt" is what
makes spacetime necessary; not the Poincare' term ("-vx/c^2") in "t ->
t - vx/c^2".

The Poincare' term did not bring about the unification of space and
time. It merely consummated a union that was already in place from the
time of Galileo.

Technically, it was an eloping of the two, rather than a fully open
marriage ceremony, since apparently nobody knew about the marriage
that Galileo (unwittingly) brought about until the time of Poincare',
Einstein and their contemporaries. But the very fact that Newton
balked at Relativity shows, all by itself, that he caught of whiff in
the air of just what space and time were doing out in the woodshed
together when nobody was looking.

Finally, the most important examples all center on the conceptual
baggage surrounding the interpretation of quantum theory, itself.
Chief amongst these is the Born Rule. This, too, is not specifically
quantum; but applies across the board to both quantum and classical
physics.

The best way to arrive at the version of the Born rule applicable to
classical systems (as well as the hybrid classico-quantum systems
where the C*-algebra has a non-trivial center) is to formulate it in
the Heisenberg picture. There was an article published in the
literature, in fact, not too long ago which did something very much
like this, though I don't have a reference handy here (do a web search
on "Classical Born Rule", this might turn up something).

[Stephen Blake]

willo_thew...@hotmail.com wrote:
> Maybe you can help with one more thing: I understand that the
> irreducible represenations are parameterized by invariants that get
> labeled things like "mass" and "spin". But...what is the rule
> that tells me which of these purely *mathematical* invariants
> get matched up with which *physical* properties? Why is the
> invariant identified with mass not identified instead with, say,
> charge?
>
The Poincare group is a semi-direct product G=TxR where T is the
Abelian subgroup of translations and R is the subgroup of generalized
rotations (spatial rotations and boosts). In order to see that the
labels for the irreps can be identified with mass, it's only necessary
to go a little way along the route to the irreps.

The irreps of an Abelian group are 1-dimensional, so the starting
point is with the subgroup T. Let vector a be a translation in T: the
1-d unitary irreps of T are,
U(a)|p,u>=exp(i<p,a>)|p,u> (1).
Here vector p labels the inequivalent 1-d irreps and <p,a> is the
scalar product in the Minkowski spacetime of which the Poincare group
is the congruence group. The label u is for copies of the irrep p; u
is for any degrees of freedom we have not identified at this stage.
Physically, p is the momentum, but we may as well define momentum as
the name for any quantity which labels the 1-d unitary irreps of an
Abelian group, so we don't need any prior notions from physics.

Let Alice and Bob be related by a generalized rotation h belonging to
subgroup R. Alice prepares a state |p,u> and Bob sees Alice's
momentum p as hp. Bob sees Alice's |p,u> as the state U(h)|p,u> which
must be some linear combination of states of momentum hp,
U(h)|p,u>=sum(v)|hp,v>Dvu(h,p) (2)
where Dvu(h,p) is some unknown matrix dependent on h,p , and with rows
and columns labelled by v and u respectively.

The response of |p,u> to a general Poincare transformation consisting
of a generalized rotation U(h) followed by a translation U(a) is,
U(a)U(h)|p,u>=exp(i<hp,a>)sum(v)|hp,v>Dvu(h,p) (3)
which is obtained by applying U(a) to (2) and using (1). Equation (3)
shows that a state |p,u> of momentum p gets moved to a state |hp,v> of
momentum hp. So, as h scans through all the elements of R, the states
of the irrep sample all the states |hp,v>. Now, the scalar product <,>
is invariant under R (and G=TxR), so,
<hp,hp>=<p,p> (4)
is the equation of a hypersurface in momentum space. So, as h scans
through the elements of R, the states of the irrep sample all the
states |hp,v> of momentum hp on the hypersurface (4). Equation (4)
just says that the hypersurface is the locus of momentum vectors which
all have the same length. So, we can label each hypersurface by the
length of the momentum vector. Finally, we can identify the label for
the irreps with mass; if a 4-momentum vector has components
(p0,p1,p2,p3), from physics, the squared length of the momentum vector
is,
p0^2-p1^2-p2^2-p3^2=m^2 (5)
where m is the rest mass. In other words, physicists call the length
of a momentum vector the rest mass.

The states |p,v> span the big Hilbert space, and the states |p,v> for
p restricted to a hypersurface of constant length <p,p>=m^2 are the
states of an irrep of the Poincare group. This subspace of states is a
particle of mass m.

Charge is a label for the irreps of the unitary group G={g=exp(it) for
t in (0,2pi)}. The group is Abelian so the irreps are 1-d and,
U(exp(it))|q>=exp(iqt)|q> (6)
for q an integer. The integers q label the irreps and the labels q
are identified with the elementary charges in suitable units.

Yours sincerely
Stephen