Subject: Re: Sophistry 103 (was: I know that!)
From: weinecks@mail1.sas.upenn.edu (Silke-Maria Weineck)
Date: 27 Oct 1996 17:24:08 GMT
Michael Zeleny (zeleny@oak.math.ucla.edu) wrote:
: weinecks@mail2.sas.upenn.edu (Silke-Maria Weineck) writes:
: >Michael Zeleny (zeleny@oak.math.ucla.edu) wrote:
: >>weinecks@mail2.sas.upenn.edu (Silke-Maria Weineck) writes:
: >>>Michael Zeleny (zeleny@oak.math.ucla.edu) wrote:
: >>>>weinecks@mail2.sas.upenn.edu (Silke-Maria Weineck) writes:
: >>>>>Michael Zeleny (zeleny@oak.math.ucla.edu) wrote:
: >>>>>>weinecks@mail1.sas.upenn.edu (Silke-Maria Weineck) writes:
: >>>>>>>Michael Zeleny (zeleny@oak.math.ucla.edu) wrote:
: >>>>>>>>weinecks@mail2.sas.upenn.edu (Silke-Maria Weineck) writes:
: >>>>>>>>>Michael Zeleny (zeleny@oak.math.ucla.edu) wrote:
: >>>>>>>>>>weinecks@mail1.sas.upenn.edu (Silke-Maria Weineck) writes:
: >>>>>>>>>>>It is very rare that physicists submit to humanities journals;
: >>>>>>>>>>>if you are suggesting that the article should have been sent
: >>>>>>>>>>>out to another physicist, I whole-heartedly agree. As things
: >>>>>>>>>>>stand, however, the hoax proves that the grad student whom
: >>>>>>>>>>>A.Ross let judge the article didn't know much about either
: >>>>>>>>>>>science or literary theory -- and what does that prove?
: >>>>>>>>>>That the postmodern "authorities", whose idiotic theses Sokal
: >>>>>>>>>>cites and purports to sustain with parodic arguments, are full
: >>>>>>>>>>of shit. Is that good enough for you?
: >>>>>>>>>No. What a silly thing to suggest. I cannot think of any
: >>>>>>>>>philosopher whose sentences cannot be made to look silly by
: >>>>>>>>>taking them out of context; when it comes to sentences spoken
: >>>>>>>>>off the record, as it were, in a matter outside their field,
: >>>>>>>>>it's so easy that only someone rather desperate for a point
: >>>>>>>>>would stoop so low. You're Erkenntnisinteresse (you understand
: >>>>>>>>>I'm using the term ironically) is running away with you.
: >>>>>>>>Your logical ineptitude is showing again. That anyone can be made to
: >>>>>>>>look stupid on the evidence of a single sufficiently decontextualized
: >>>>>>>>quotation, does not entail that no single quotation can serve as a
: >>>>>>>>sufficient proof of its author's stupidity, as witness le "sottisier"
: >>>>>>>>de Bouvard et Pécuchet. In effect, Sokal does for the postmodern
: >>>>>>>>booboisie what Flaubert did for the French bourgeoisie.
: >>>>>>>Nonsense. Sokal's hoax proves exactly nothing about Derrida. It proves
: >>>>>>>something about Social Text. Your work isn't done, and if you rely on
: >>>>>>>the likes of Sokal to do it, you are wimping out.
: >>>>>>Derrida's assertion cited by Sokal, about the Einsteinian constant not
: >>>>>>being a constant, proves two things. Firstly, he is ageometretos, and
: >>>>>>hence not a philosopher. Secondly, he is given to pronouncing on the
: >>>>>>basis of ignorance, and hence not a critic. Why would you doubt that?
: >>>>>Simple. It does not follow, and you haven't produced an argument. A) you
: >>>>>have no idea what he meant. B) Even if you had an idea what he meant and
: >>>>>even if your idea were correct, it wouldn't follow that he's not a
: >>>>>philosopher, since "philosophy" is not defined as "that body of work that
: >>>>>exhibits knowledge of Einstein." C) a critic can be ignorant of many
: >>>>>things he pronounces on, as long as he doesn't pronounce on them _qua_
: >>>>>critic in his field.
: >>>>Here is an argument. A) I have a good idea what Einstein meant, and
: >>>>an equally good idea that any reasonable interpretation of Derrida's
: >>>>comment is incompatible with Einstein's meaning.
: >>>Please share your insight, then.
: >>It is not insight, but learning, which is not something I can share with
: >>a passive audience. I recommend the Feynman Lectures as a good starting
: >>point in this matter.
: >Feynman explains what Hippolite meant? That's fascinating. Are you sure?
: Are you suffering from ADD? The point is to address Einstein's meaning.
The point is that you can only determine whether Derrida's meaning is
incompatible with Einstein's meaning if you know what Derrida's meaning
is. Elementary logic. So do you or don't you have an interpretation of
what either Hippolite or Deridda were saying?
: >>>> B) Since Derrida
: >>>>aims to debunk Platonism, since the understanding of Platonism depends
: >>>>on the understanding of geometry, and since Einstein is the wellspring
: >>>>of modern geometry, Derrida's ignorance automatically condemns his
: >>>>project to failure.
: >>>This is fun, but it's not an argument.
: >>It is an argument, and a logically valid one. Under the circumstances,
: >>I would be willing to let frêre Jacques off the hook if only he had
: >>evinced minimal acquaintance with Euclid, never mind Lobachevsky or
: >>Riemann. Alas, it is not forthcoming.
: >Logical, hm. You would have to prove that
: >a) an understanding of Platonism does indeed depend on an understanding
: >of geometry; wild assertion no. 1
: >b) that Einstein is indeed Platonism for the 20th century -- here, you'd
: >run into trouble with Russell Turpin and others
: >c) that Derrida "debunks" Platonism or intends to do so -- here, we'd
: >only need a quote, so you wouldn't have to think much
: >d) that all continental philosophy after Kant is cognizant of modern math.
: In order to prove a), I need do no more than invoke the principle of
: interpretive charity -- if Plato says that an ageometretos must be
: disqualified, he is to be taken at his word, so long as your aim is to
: understand Plato. To address b) independently of your predictably
: inaccurate reading, I need not do any more than restate the point:
: Einstein's work on the geometry of the space-time manifold yields the
: basis of present-day understanding of physical geometry. Curiously
: enough, I have already satisfied c) in my conversation with Brian
: Artese by quoting Derrida and citing Plato to the contrary. As
: regards d), to adapt your own idiom, you are confusing an intellectual
: enterprise with its institutionalized idiots.
a) qualifications for philosophers vary greatly within Plato's oeuvre. I
can understand that you would pick the one agreeable with your
Erkenntnisinteresse. However, if you were to direct your attention to the
Phaedrus, you will see a different definition of philosopher emerge.
b) is irrelevant
c) a disagreement does not constitute a debunking
d) on the contrary, that is your domain. Are you disqualifying all
philosophers from philosophy who have not proven to you that they know
all they could about geometry, yes or no?
: >>Do you think it is a coincidence that the best portrayal of postmodern
: >>criticism to date was presented by Nabokov as early as 1962?
: >I would find it highly surprising, yes. I never thought Nabokov was a
: >good critic, btw, even though he's a great writer. Like Kleist or
: >Buechner in that regard.
: Excellence of portrayal is a matter of strength and precision in
: observation and representation, rather than of the critical virtues
: of analytic meticulousness in interpretation and explanation.
That is hardly a response. You have yet to bolster your claim as to what
Nabokov achieved.
: >The image
: >>of a logorrhetic, vituperative, frustrated uranist, equally ignorant of
: >>Euclid and Shakespeare, may fit Barthes a little bit better than it does
: >>Derrida. Then again -- I know not what really turns on the eminent
: >>grand-daddy of decon.
: >How odd. Barthes of all people, "vituperative," "frustrated"? That most
: >loving of critics?
: As loving as Charles Kinbote, and equally preoccupied with conspiring
: against the author. As regards his frustration, it is my understanding
: that fulfillment does not conduce to throwing oneself under a truck.
As far as I know, he didn't "throw himself under a truck," but was hit by
a truck. Loving literature is not the same as loving the 18th century
conception of authorship.
: >>>> C) The copyright laws imply that any critical
: >>>>comments appearing in print of symposium proceedings are subject to
: >>>>the speaker's release of publication rights and hence carry the
: >>>>presumption of ex cathedra pronouncements.
: >>>Perhaps they do; that such is enforced, is, however, amply disproven.
: >>>Just witness Wolin's mistranslation of Derrida and subsequent publication.
: >>On the basis of personal experience with intellectual property laws, I
: >>assure you that such enforcement by the owner is always possible among
: >>the signatories to the Berne convention.
: >For someone harping on logic, this is quite below par. It might be
: >possible to enforce; that does not imply that it has been enforced. So?
: So if Derrida had been cognizant of revealing that he was full of shit,
: he would not have allowed published dissemination of this revelation.
He might give his readers more credit than they deserve. Then again, he
might only be interested in those readers who do deserve minimal credit.
: >>>>>>As you know, I have done my work and need not rely on Sokal to do it.
: >>>>>>Nonetheless, if I wanted to cite a professional opinion that Derrida
: >>>>>>was a charlatan, I would have brought up Chomsky.
: >>>>>I don't know this at all. I'm still waiting for you to exhibit a
: >>>>>rudimentary understanding of Derrida's argument in "Cogito." As long as
: >>>>>you can't tell us what it is you object to, your objections won't be
: >>>>>taken seriously.
: >>>>In the beginning of our exchange I told you the rules of engagement --
: >>>>each thrust is to be followed by a parry and vice versa. By continuing
: >>>>to argue, you implicitly accepted the conventional rules. If you wish
: >>>>to make a request, I will consider it after you reply to my last article
: >>>>point by point.
: >>>The last exchange failed. A reasonable reaction to failure is to try
: >>>something else.
: >>I will reasonably consider trying something else after you reply to my
: >>last article point by point.
: >In other words, you're wimping out?
: In my opinion, true wimpery is exemplified by the party who excused
: herself from following the rules she had accepted from the start.
You are, of course, entitled to your opinions. Let us all agree to note,
though, that you refused to exhibit understanding of an essay you claim
to have read and you claim to object to on philosophical grounds; your
reasons may be what you say they are.
Silke
Subject: This Week's Finds in Mathematical Physics (Week 93)
From: baez@math.ucr.edu (John Baez)
Date: 27 Oct 1996 18:17:25 -0800
This Week's Finds in Mathematical Physics - Week 93
John Baez
Lately I've been trying to learn more about string theory. I've always
had grave doubts about string theory, but it seems worth knowing about.
As usual, when I'm trying to learn something I find it helpful to write
about it --- it helps me remember stuff, and it points out gaps in my
understanding. So I'll start trying to explain some string theory in
this and forthcoming Week's Finds.
However: watch out! This isn't going to be a systematic introduction to
the subject. First of all, I don't know enough to do that. Secondly,
it will be very quirky and idiosyncratic, because the aspects of string
theory I'm interested in now aren't necessarily the ones most string
theorists would consider central. I've been taking as my theme of
departure, "What's so great about 10 and 26 dimensions?" When one
reads about string theory, one often hears that it only works in 10
or 26 dimensions --- and the obvious question is, why?
This question leads one down strange roads, and one runs into lots of
surprising coincidences, and spooky things that sound like coindences
but might NOT be coincidences if we understood them better.
For example, when we have a string in 26 dimensions we can think of it
as wiggling around in the 24 directions perpendicular to the
2-dimensional surface the string traces out in spacetime (the "string
worldsheet"). So the number 24 plays an especially important role in
26-dimensional string theory. It turns out that
1^2 + 2^2 + 3^2 + ... + 24^2 = 70^2.
In fact, 24 is the only* integer n > 1 such that the sum of squares
from 1^2 to n^2 is itself a perfect square. Is this a coincidence?
Probably not, as I'll eventually explain! This is just one of
many eerie facts one meets when trying to understand this stuff.
For starters I just want to explain why dimensions of the form 8k + 2
are special. Notice that if we take k = 0 here we get 2, the
dimension of the string worldsheet. For k = 1 we get 10, the dimension
of spacetime in "supersymmetric string theory". For k = 3 we get 26,
the dimension of spacetime in "purely bosonic string theory". So these
dimensions are important. What about n = k and the dimension 18, I hear
you ask? Well, I don't know what happens there yet... maybe someone can
tell me! All I want to do now is to explain what's good about
8n + 2.
But I need to start by saying a bit about fermions.
Remember that in the Standard Model of particle physics --- the model
that all fancier theories are trying to outdo --- elementary particles
come in 3 basic kinds. There are the basic fermions. In general a
"fermion" is a particle whose angular momentum comes in units of
Planck's constant hbar times 1/2, 3/2, 5/2, and so on. Fermions satisfy
the Pauli exclusion principle --- you can't put two identical fermions
in the same state. That's why we have chemistry: the electrons stack up
in "shells" at different energy levels, instead of all going to the
lowest-energy state, because they are fermions and satisfy the exclusion
principle. In the Standard Model the fermions go like this:
LEPTONS QUARKS
electron electron neutrino up quark down quark
muon muon neutrino strange quark charm quark
tauon tauon neutrino top quark bottom quark
There are three "generations" here, all rather similar to each other.
There are also particles in the Standard Model called "bosons" having
angular momentum in units of hbar times 0,1,2, and so on. Identical
bosons, far from satisfying the exclusion principle, sort of like to all
get into the same state: one sees this in phenomena such as lasers,
where lots of photons occupy the same few states. Most of the bosons
in the Standard Model are called "gauge bosons". These carry the
different forces in the standard model, by which the particles interact:
ELECTROMAGNETIC FORCE WEAK FORCE STRONG FORCE
photon W+ 6 gluons
W-
Z
Finally, there is also a bizarre particle in the Standard Model called the
"Higgs boson". This was first introduced as a rather ad hoc hypothesis:
it's supposed to interact with the forces in such a way as to break the
symmetry that would otherwise be present between the electromagnetic
force and the weak force. It has not yet been observed; finding it would
would represent a great triumph for the Standard Model, while *not*
finding it might point the way to better theories.
Indeed, while the Standard Model has passed many stringent experimental
tests, and successfully predicted the existence of many particles which
were later observed (like the W, the Z, and the charm and top quarks),
it is a most puzzling sort of hodgepodge. Could nature really be this
baroque at its most fundamental level? Few people seem to think so;
most hope for some deeper, simpler theory.
It's easy to want a "deeper, simpler theory", but how to get it? What
are the clues? What can we do? Experimentalists certainly have their
work cut out for them. They can try to find or rule out the Higgs.
They can also try to see if neutrinos, assumed to be massless in the
Standard Model, actually have a small mass --- for while the Standard
Model could easily be patched if this were the case, it would shed
interesting light on one of the biggest mysteries in physics, namely why
the fermions in nature seem not to be symmetric under reflection, or
"parity". Right now, we believe that neutrinos only exist in a
left-handed form, rotating one way but not the other around the axis
they move along. This is intimately related to their apparent
masslessness. In fact, for reasons that would take a while to explain,
the lack of parity symmetry in the Standard Model forces us to assume
all fermions acquire their mass only through interaction with the Higgs
particle! For more on the neutrino mass puzzle, try:
1) Paul Langacker, Implications of neutrino mass,
http://dept.physics.upenn.edu/~www/neutrino/jhu/jhu.html
And, of course, experimentalists can continue to do what they always do
best: discover the utterly unexpected.
Theorists, on the other hand, have been spending the last couple of
decades poring over the standard model and trying to understand what
it's telling us. It's so full of suggestive patterns and partial
symmetries! First, why are there 3 forces here? Each force goes along
with a group of symmetries called a "gauge group", and electromagnetism
corresponds to U(1), while the weak force corresponds to SU(2) and the
strong force corresponds to SU(3). (Here U(n) is the group of n x n
unitary complex matrices, while SU(n) is the subgroup consisting of
those with determinant equal to 1.) Well, actually the Standard Model
partially unifies the electromagnetic and weak force into the
"electroweak force", and then resorts to the Higgs to explain why these
forces are so different in practice. Various "grand unified theories"
or "GUTs" try to unify the forces further by sticking the group SU(3) x
SU(2) x U(1) into a bigger group --- but then resort to still more
Higgses to break the symmetry between them!
Then, there is the curious parallel between the leptons and quarks in
each generation. Each generation has a lepton with mass, a massless or
almost massless neutrino, and two quarks. The massive lepton has charge
-1, the neutrino has charge 0 as its name suggests, the "down" type
quark has charge -1/3, and the "up" type quark has charge 2/3. Funny
pattern, eh? The Standard Model does not really explain this, although
it would be ruined by "anomalies" --- certain nightmarish problems that
can beset a quantum field theory --- if one idly tried to mess with the
generations by leaving out a quark or the like. Indeed, this is why the
charm quark was first predicted, before the generation pattern was fully
apparent. It's natural to try to "unify" the quarks and leptons, and
indeed, in grand unified theories like the SU(5) theory proposed in 1974
of Georgi and Glashow, the quarks and leptons are treated in a unified
way.
Another interesting pattern is the repetition of generations itself.
Why is there more than one? Why are there three, almost the same,
but with the masses increasing dramatically as we go up? The Standard
Model makes no attempt to explain this, although it does suggest that
there had better be more than 17 quarks --- more, and the strong force
would not be "asymptotically free" (weak at high energies), which would
cause lots of problems for the theory. In fact, experiments strongly
suggest that there are no more than 3 generations. Why?
Finally, there is the grand distinction between bosons and fermions.
What does this mean? Here we understand quite a bit from basic
principles. For example, the "spin-statistics theorem" explains why
particles with half-integer spin should satisfy the Pauli exclusion
principle, while those with integer spin should like to hang out
together. This is a very beautiful result with a deep connection to
topology, which I try to explain in
2) John Baez, Spin, statistics, CPT and all that jazz,
http://math.ucr.edu/home/baez/spin.stat.html
But many people have tried to bridge the chasm between bosons and
fermions, unifying them by a principle called "supersymmetry". As in
the other cases mentioned above, when they do this, they then need to
pull tricks to "break" the symmetry to get a theory that fits the
experimental fact that bosons and fermions are very different.
Personally, I'm suspicious of all these symmetries postulated only to be
cleverly broken; this approach was so successful in dealing with the
electroweak force --- modulo the missing Higgs! --- that it seems to
have been accepted as a universal method of having ones cake and eating
it too.
Now, string theory comes in two basic flavors. Purely bosonic
string theory lives in 26 dimensions and doesn't have any fermions in
it. Supersymmetric string theories live in 10 dimensions and have both
bosons and fermions, unified via supersymmetry. To deal with the
fermions in nature, most work in physics has focused on the
supersymmetric case. Just for completeness, I should point out that
there are 5 different supersymmetric string theories: type I, type
IIA, type IIB, E8 x E8 heterotic and SO(32) heterotic. For more on
these, see "week72". We won't be getting into them here. Instead,
I just want to explain how fermions work in different dimensions, and
why nice things happen in dimensions of the form 8k + 2. Most of
what I say is in Section 3 of
3) John H. Schwarz, Introduction to supersymmetry, in Superstrings
and Supergravity, Proc. of the 28th Scottish Universities Summer
School in Physics, ed. A. T. Davies and D. G. Sutherland, University
Printing House, Oxford, 1985.
but mathematicians may also want to supplement this with material
from the book "Spin Geometry" by Lawson and Michelson, cited in
"week82".
To understand fermions in different dimensions we need to understand
Clifford algebras. As far as I know, when Clifford originally invented
these algebras in the late 1800s, he was trying to generalize Hamilton's
quaternion algebra by considering algebras that had lots of different
anticommuting square roots of -1. In other words, he considered
an associative algebra generated by a bunch of guys e_1,...,e_n,
satisfying
e_i^2 = -1
for all i, and
e_i e_j = - e_j e_i
whenever i is not equal to j. I discussed these algebras in "week82"
and I said what they all were --- they all have nice descriptions in terms
of the reals, the complexes, and the quaternions.
These original Clifford algebras are great for studying rotations in
n-dimensional Euclidean space --- please take my word for this for now.
However, here we want to study rotations and Lorentz transformations
in n-dimensional Minkowski spacetime, so we need to work with a slightly
Different kind of Clifford algebra, which was probably invented by Dirac.
In n-dimensional Euclidean space the metric (used for measuring distances)
is
dx_1^2 + dx_2^2 + ... + dx_n^2
while in n-dimensional Minkowski spacetime it is
dx_1^2 + dx_2^2 + ... - dx_n^2
or if you prefer (it's just a matter of convention), you can
take it to be
- dx_1^2 - dx_2^2 - ... + dx_n^2
So it turns out that we need to switch some signs in the definition
of the Clifford algebra when working in Minkowski spacetime.
In general, we can define the Clifford algebra C_{p,q} to be the algebra
generated by a bunch of elements e_i, with p of them being square roots
of -1 and q of them being square roots of 1. As before, we require that
they anticommute:
e_i e_j = - e_j e_i
when i and j are different. Physicists usually call these guys "gamma
matrices". For n-dimensional Minkowski space we can work either
with C_{n-1,1} or C_{1,n-1}, depending on our preference. As Cecile
DeWitt has pointed out, it *does* make a difference which one we use.
With some work, one can check that these algebras go like this:
C_{0,1} R + R C_{1,0} C
C_{1,1} R(2) C_{1,1} R(2)
C_{2,1} C(2) C_{1,2} R(2) + R(2)
C_{3,1} H(2) C_{1,3} R(4)
C_{4,1} H(2) + H(2) C_{1,4} C(4)
C_{5,1} H(4) C_{1,5} H(4)
C_{6,1} C(8) C_{1,6} H(4) + H(4)
C_{7,1} R(16) C_{1,7} H(8)
I've only listed these up to 8-dimensional Minkowski spacetime, and
the cool thing is that after that they sort of repeat --- more precisely,
C_{n+8,1} is just the same as 16 x 16 matrices with entries in C_{n,1},
and C_{1,n+8} is just 16 x 16 matrices with entries in C_{n,1}!
This "period-8" phenomenon, sometimes called Bott periodicity, has
implications for all sorts of branches of math and physics. This is
why fermions in 2 dimensions are a bit like fermions in 10 dimensions
and 18 dimensions and 26 dimensions....
In physics, we describe fermions using "spinors", but there are
different kinds of spinors: Dirac spinors, Weyl spinors, Majorana
spinors, and even Majorana-Weyl spinors. This is a bit technical but
I want to dig into it here, since it explains what's special about
8k + 2 dimensions and especially 10 dimensions.
Before I get technical, though, let me just summarize the point for
those of you who don't want all the gory details. "Dirac spinors"
are what you use to describe spin-1/2 particles that come in both
left-handed and right-handed forms and aren't their own antiparticle
--- like the electron. Weyl spinors have half as many components,
and describe spin-1/2 particles with an intrinsic handedness that
aren't their own antiparticle --- like the neutrino. "Weyl spinors"
are only possible in even dimensions!
Both these sorts of spinors are "complex" --- they have complex-valued
components. But there are also real spinors. These are used for describing
particles that are their own antiparticle, because the operation of
turning a particle into an antiparticle is described mathematically
by complex conjugation. "Majorana spinors" describe spin-1/2 particles
that come in both left-handed and right-handed forms and are their
own antiparticle. Finally, "Majorana-Weyl spinors" are used to describe
spin-1/2 particles with an intrinsic handedness that are their own
antiparticle.
As far as we can tell, none of the particles we've seen are Majorana
or Majorana-Weyl spinors, although if the neutrino has a mass it
might be a Majorana spinor. Majorana and Majorana-Weyl spinors
only exist in certain dimensions. In particular, Majorana-Weyl spinors
are very finicky: they only work in dimensions of the form 8k + 2.
This is part of what makes supersymmetric string theory work in 10
dimensions!
Now let me describe the technical details. I'm doing this mainly
for my own benefit; if I write this up, I'll be able to refer to
it whenever I forget it. For those of you who stick with me, there
will be a little reward: we'll see that a certain kind of supersymmetric
gauge theory, in which there's a symmetry between gauge bosons and
fermions, only works in dimensions 3, 4, 6, and 10. Perhaps
coincidentally --- I don't understand this stuff well enough to know ---
these are also the dimensions when supersymmetric string theory works
classically. (It's the quantum version that only works in dimension 10.)
So: part of the point of these Clifford algebras is that they give
representations of the double cover of the Lorentz group in different
dimensions. In "week61" I explained this double cover business,
and how the group SO(n) of rotations of n-dimensional Euclidean space
has a double cover called Spin(n). Similarly, the Lorentz group
of n-dimensional Minkowski space, written SO(n-1,1), has a double cover
we could call Spin(n-1,1). The spinors we'll discuss are all
representations of this group.
The way Clifford algebras help is that there is a nice way to
embed Spin(n-1,1) in either C_{n-1,1} or C_{1,n-1}, so any
representation of these Clifford algebras gives a representation
of Spin(n-1,1). We have a choice of dealing with real representations or
complex representations. Any complex representation of one of
these Clifford algebras is also a representation of the *complexified*
Clifford algebra. What I mean is this: above I implicitly wanted
C_{p,q} to consist of all *real* linear combinations of products of
the e_i, but we could have worked with *complex* linear combinations
instead. Then we would have "complexified" C_{p,q}. Since the
complex numbers include a square root of minus 1, the complexification
of C_{p,q} only depends on the dimension p + q, not on how many minus
signs we have.
Now, it is easy and fun and important to check that if you complexify R
you get C, and if you complexify C you get C + C, and if you complexify
H you get C(2). Thus from the above table we get this table:
dimension n complexified Clifford algebra
1 C + C
2 C(2)
3 C(2) + C(2)
4 C(4)
5 C(4) + C(4)
6 C(8)
7 C(8) + C(8)
8 C(16)
Notice this table is a lot simpler --- complex Clifford algebras
are "period-2" instead of period-8.
Now the smallest complex representation of the complexified Clifford
algebra in dimension n is what we call a "Dirac spinor". We can figure
out what this is using the above table, since the smallest complex
representation of C(n) or C(n) + C(n) is on the n-dimensional complex
vector space C^n, given by matrix multiplication. Of course, for
C(n) + C(n) there are *two* representations depending on which copy
of C(n) we use, but these give equivalent representations of Spin(n-1,1),
which is what we're really interested in, so we still speak of "the"
Dirac spinors.
So we get:
dimension n Dirac spinors
1 C
2 C^2
3 C^2
4 C^4
5 C^4
6 C^8
7 C^8
8 C^16
The dimension of the Dirac spinors doubles as we go to each new
even dimension.
We can also look for the smallest real representation of C_{n-1,1}
or C_{1,n-1}. This is easy to work out from our tables using
the fact that the algebra R has its smallest real representation
on R, while for C it's on R^2 and for H it's on R^4.
Sometimes this smallest real representation is secretly just the
Dirac spinors *viewed as a real representation* --- we can view C^n
as the real vector space R^{2n}. But sometimes the Dirac spinors
are the *complexification* of the smallest real representation ---
for example, C^{2n} is the complexification of R^n. In this
case folks call the smallest real representation "Majorana spinors".
When we are looking for the smallest real representations, we get
different answers for C_{n-1,1} and C_{1,n-1}. Here is what we get:
n C_{n-1,1} smallest C_{1,n-1} smallest
real rep real rep
1 R + R R Majorana C R^2
2 R(2) R^2 Majorana R(2) R^2 Majorana
3 C(2) R^4 R(2) + R(2) R^2 Majorana
4 H(2) R^8 R(4) R^4 Majorana
5 H(2) + H(2) R^8 C(4) R^8
6 H(4) R^16 H(4) R^16
7 C(8) R^16 H(4) + H(4) R^16
8 R(16) R^16 Majorana H(8) R^32
I've noted when the representations are Majorana spinors. Everything
repeats with period 8 after this, in an obvious way.
Finally, sometimes there are "Weyl spinors" or "Majorana-Weyl"
spinors. The point is that sometimes the Dirac spinors, or
Majorana spinors, are a *reducible* representation of Spin(1,n-1).
For Dirac spinors this happens in every even dimension, because the
Clifford algebra element Gamma = e_1 ... e_n commutes with everything
in Spin(1,n-1) and Gamma^2 is 1 or -1, so we can break the space of
Dirac spinors into the two eigenspaces of Gamma, which will be smaller
reps of Spin(1,n-1) --- the "Weyl spinors". Physicists usually call this
Gamma thing "gamma_5", and it's an operator that represents parity
transformations. We get "Majorana-Weyl" spinors only when we have
Majorana spinors, n is even, and Gamma^2 = 1, since we are then working
with real numbers and -1 doesn't have a square root. You can work out
Gamma^2 for either and C_{n-1,1} or C_{1,n-1}, and see that we'll
only get Majorana-Weyl spinors when n = 8k + 2.
Whew! Let me summarize some of our results:
n Dirac Majorana Weyl Majorana-Weyl
1 C R
2 C^2 R^2 C R
3 C^2 R^2
4 C^4 R^4 C^2
5 C^4
6 C^8 C^4
7 C^8
8 C^16 R^16 C^8
When there are blanks here, the relevant sort of spinor doesn't
exist. Here I'm not distinguishing Majorana spinors that come from
C_{n-1,1} and those that come from C_{1,n-1}; you can do that with
the previous table. Again, things continue for larger n in an obvious
way.
Now, let's imagine a theory that has a supersymmetry between a gauge
bosons and a fermion. We'll assume there are as many physical degrees of
freedom for the gauge boson as there are for the fermion. Gauge
bosons have n - 2 physical degrees of freedom in n dimensions: for
example, in dimension 4 the photon has 2 degrees of freedom, the spin-up
and the spin-down states. So we want to find a kind of spinor that
has n - 2 physical degrees of freedom. But the number of physical
degrees of freedom of a spinor field is half the number of (real)
components of the spinor, since the Dirac equation relates the
components. So we are looking for a kind of spinor that has 2(n - 2)
real components. This occurs in only 4 cases:
n = 3 -> 2(n-2) = 2, and Majorana or Weyl spinors have 2 real components
n = 4 -> 2(n-2) = 4, and Majorana or Weyl spinors have 4 real components
n = 6 -> 2(n-2) = 8, and Weyl spinors have 8 real components
n = 10 -> 2(n-2) = 16, and Majorana-Weyl spinors have 16 real components
Note we count complex components as two real components. And note how
dimension 10 works: the dimension of the spinors grows pretty fast as
n increases, but the Majorana-Weyl condition reduces the dimension by
a factor of 4, so dimension 10 just squeaks by!
John Schwarz explains how nice things happen in the same dimensions
for superstring theory in:
4) John H. Schwarz, Introduction to superstrings, in Superstrings
and Supergravity, Proc. of the 28th Scottish Universities Summer
School in Physics, ed. A. T. Davies and D. G. Sutherland, University
Printing House, Oxford, 1985.
He also makes a tantalizing remark: perhaps these 4 cases correspond
somehow to the reals, complexes, quaternions and octonions. Note:
3 = 1 + 2, 4 = 2 + 2, 6 = 4 + 2 and 10 = 8 + 2. You can never tell
with this stuff... everything is related.
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