Dirac Gamma matrices including gamma^5, and the Spacetime Metric g_uv
Jay R. Yablon
It is of course well-known that the (contravariant) spacetime metric
g^uv and Dirac gamma matrices are related by the commutation
relationship:
g^uv = (1/2) (gamma^u gamma^v + gamma^v gamma^u) (1)
where u = 0,1,2,3 are spacetime indices. We recognize that only where
g^uv = n^uv (the Minkowski metric) will the gamma^u be precisely equal
to the well-known Dirac matrices which incorporate pairs of the Pauli
matrices. Where g^uv = n^uv, the metric is diagonal with diag (g_uv) =
(1,-1,-1,-1). Otherwise, where g^uv not= n^uv, the Dirac matrices
themselves ought to vary as well (yes? no?).
I like to think of this relationship (1) as saying that the Dirac
equation is the "operator square root" of the metric equation for the
spacetime interval ds, because one can start with the metric equation:
dtau^2 = g^uv dx_u dx_v (2)
with g^uv given by (1), separate this into the (duplicated) equation:
dtau = gamma^u dx_u (3)
and then used this to operate on a four-component Dirac spinor psi in
the form:
dtau psi = (gamma^u dx_u)psi (4)
Subtracting dtau psi from each side, multiplying through by a mass m,
and dividing through by dtau, with the four momentum defined as p_u = m
(dx_u/dtau), then yields:
0 = (gamma^u p_u - m) psi (5)
which is Dirac's equation in classical form. The road to quantum
mechanics then runs through p_u --> iD_u, with the gauge-covariant
derivative D_u bringing in gauge fields.
Here are my questions:
1) Given equation (1), is it fair to think of the gamma^u as being just
as fundamental to the structure of spacetime as the g^uv, and perhaps
even more so because the gamma^u have certain features (such as their
being able to accommodate Dirac spinors which the g_uv alone cannot do,
and the axial gamma^5 matrix) which are not at all apparent just looking
at g^uv? Put differently, in general relativity we define spacetime by
its metric. Can we equally think, and maybe even more fundamentally
think, that spacetime is really defined by its gamma matrices? In other
words, can we think of the Dirac gammas as the "structure matrices of
spacetime" which, via (1), define a classical metric?
2) If the Dirac gammas can be thought of as the "structure matrices of
spacetime," then can we also think of the axial gamma^5 as a fifth
structure matrix of spacetime?
3) Would it make sense to rewrite (1), including the gamma^5, as:
g^UV = (1/2) (gamma^U gamma^V + gamma^V gamma^U) (6)
with where U = 0,1,2,3,5?
4) With U=5, does it make sense to conclude that the existence of the
gamma^5 is indicative of a fifth spacetime dimension?
5) Wherever the gamma^U are taken to be the Dirac matrices
incorporating pairs of Pauli matrices, the (Minkowskian) metric defined
by (6) then has diag (g_UV) = (1,-1,-1,-1,1). This gives this "fifth"
dimension a timelike signature. Does it make any sense, therefore, to
think of this fifth dimension originating in gamma^5 as a second, "axial
time" dimension? (Which would lead then to being able to "rotate"
between the ^0 and ^5 time dimension leading to a many-fingered time
sort of notion which I recall Feynman once entertained.)
I am looking for any flaws you can identify in this line of thought,
including whatever is the "conventional wisdom" on having more than one
timelike dimension.
Thanks,
Jay.
_____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
Igor Khavkine
> Dear SPR friends,
>
> It is of course well-known that the (contravariant) spacetime metric
> g^uv and Dirac gamma matrices are related by the commutation
> relationship:
>
> g^uv = (1/2) (gamma^u gamma^v + gamma^v gamma^u) (1)
>
> where u = 0,1,2,3 are spacetime indices.
> 1) Given equation (1), is it fair to think of the gamma^u as being just
> as fundamental to the structure of spacetime as the g^uv, and perhaps
> even more so because the gamma^u have certain features (such as their
> being able to accommodate Dirac spinors which the g_uv alone cannot do,
> and the axial gamma^5 matrix) which are not at all apparent just looking
> at g^uv? Put differently, in general relativity we define spacetime by
> its metric. Can we equally think, and maybe even more fundamentally
> think, that spacetime is really defined by its gamma matrices? In other
> words, can we think of the Dirac gammas as the "structure matrices of
> spacetime" which, via (1), define a classical metric?
Whether something is "fundamental" is often a very ill-posed question.
A better question is: how does one construct gamma matrices and why are
they important? Once this question is answered, the rest of these
questions will evaporate.
[...stuff about fith dimension...]
> I am looking for any flaws you can identify in this line of thought,
> including whatever is the "conventional wisdom" on having more than one
> timelike dimension.
The flaw is a common one, it is blindly going by analogy.
Igor
Jay R. Yablon
of thought," I was hoping for a serious response.
Yes, the gamma^5 are staring us in the face and someone naiive could
just say, "gee, that ought to be a fifth spacetime dimension" without
more than superficial analysis, and I would then agree with the comments
made. However, let me elaborate my questions, please, because the
mathematics hangs together quite well and to not seriously consider this
and give a dismissive answer to me reflects a prejudice in thinking.
Here are my more detailed questions, which I do not think will
evaporate or leave the room:
1) Am I correct to understand that when equation (1) is written with
g^uv and not with n^uv, that this is done deliberately with the
understanding that g_uv could be / is the contravariant metric tensor
g^uv = n^uv + kappa h^uv, where h^uv represents a non-zero gravitational
field?
2) Am I correct therefore, in thinking that in a gravitational field,
each of the Dirac matrices themselves might differ from aforementioned
the pairs of Pauli matrices as well, by some 4x4 spinor matrix h^u? In
other words, is to correct to think that in a gravitational field, one
may write something of the form:
gamma^u = n^u + K h^u, (2)
where n^u are the usual gamma^u utilizing Pauli matrix pairs and this
difference h^u is effectively another way of representing the
gravitational field h^uv within the gamma^u, with K being some constant
related to kappa?
3) Given equation (1), if the g^uv are in fact the g^uv = n^uv + kappa
h^uv which define the geometric curvature of spacetime, is it fair to
think of the gamma^u = n^u + K h^u as being just as capable of
representing spacetime curvature as g^uv = n^uv + kappa h^uv, and even
further, as capturing certain subtleties in spacetime structure because
of their spinor nature which the g^uv cannot alone capture? Put
differently, in general relativity we define the curvature of spacetime
by its metric. Can we equally think, that spacetime is alternatively
defined by its gamma matrices gamma^u, from which the g^uv may in turn
be deduced by (1)? In other words, can we think of the Dirac gammas as
the "structure matrices of spacetime" which, via (1), give us an
alternative way to define a classical spacetime metric?
4) The axial gamma^5 matrix is defined by gamma^5 = i gamma^0 gamma^1
gamma^2 gamma^3, and in this way, looks to be a "peer" of all the other
gamma^u. For example, we can equally write gamma^0 = i gamma^1 gamma^2
gamma^3 gamma^5, or gamma^1 = i gamma^0 gamma^2 gamma^3 gamma^5 or
gamma^3 = i gamma^0 gamma^1 gamma^2 gamma^5 or gamma^2 = -i gamma^0
gamma^1 gamma^3 gamma^5 (note that gamma^2 is the only matrix which
flips the sign when isolated). If the Dirac gamma^u can be thought of
as the "structure matrices of spacetime," and because gamma^5 appears to
be on a completely equal footing with the gamma^u, with u = 0,1,2,3, at
least via the relationship:
1 = i gamma^0 gamma^1 gamma^2 gamma^3 gamma^5 (3)
is there any reason why we cannot also think of the axial gamma^5 as a
fifth structure matrix of spacetime?
5) Is there any reason, therefore, why we ought not rewrite the
spacetime metric tensor (1), including the gamma^5, as:
g^UV = (1/2) (gamma^U gamma^V + gamma^V gamma^U) (4)
with U = 0,1,2,3 and 5?
6) If we can fairly define a metric tensor g^UV with U = 0,1,2,3 and 5,
as in equation (4), does it make sense to conclude that the existence of
the gamma^5 is indicative of a fifth spacetime dimension? IF NOT, WHY
NOT?
7) Wherever the gamma^U are taken to be the Dirac matrices
incorporating pairs of Pauli matrices, the (Minkowskian) metric g_UV =
n_UV defined by (4) then has:
diag (g_UV=n_UV) = (1,-1,-1,-1,1). (5)
This "fifth" dimension thereby naturally acquires a timelike signature.
Does it make any sense, therefore, to think of this fifth dimension
originating in gamma^5 as a second, "axial time" dimension?
8) If gamma^5 signifies a second time dimension, which yields a 5 = d =
D + 2 dimensional spacetime with D=3 space dimensions, then is there any
reason why we cannot define an invariant "mega-"proper time differential
interval dT in addition to the usual proper time invariant dtau, and use
g_UV to raise and lower indexes, according to:
dT^2 = g_UV dx^U dx^V =dx^U dx_U
= dx^u dx_u + dx^5 dx_5 = dtau^2 + dx^5 dx_5 ? (6)
9) Then just as we can perform ordinary rotations through the D=3 space
dimensions, is there anything which would bar us from considering a
rotation between the usual time dimension x^0 and an axial time
dimension x^5? Then, just as Feynman taught that particles, e.g.,
electrons can move "backwards" through time, might we also "define" the
reference frame of an observer as one in which dx_5=0, always, and
consider some particles as moving with a "sideways" component through
time, dx^5 not=0 relative an observer's movement through time? For
example, might we even think of a massless photon or graviton, for which
dtau=0, as moving fully sideways through time, with dT = dx^5 but
dtau=0?
10) Does this lead, at least roughly, to a "many-fingered" time sort of
notion which I recall Feynman once entertained? What is the
modern"conventional wisdom" and what other viewpoints are there on such
things as having more than one timelike dimension, e.g., two timelike
dimensions, including references which address this point? Has anyone
ever examined what quantum field theory would look like with a second
time dimension, that is, has anyone ever explored d = D + 2 QFT, for D=3
and otherwise? If so, where might I find such examination?
Please reply carefully. The fact that the analogy may be staring us in
the face does not mean the analogy is wrong. There seems to be little
question that one can write down equation (4),
g^UV = (1/2) (gamma^U gamma^V + gamma^V gamma^U).
and that at least the subset g^uv is recognized to define the usual
spacetime metric. The question is whether there is some reason *why*
one cannot think of g^u5 and g^55 as extended components of an extended
spacetime metric where one then proceeds to examine symmetries not only
in the 4-dimensional spacetime but also in the 5-dimensional manifold.
For example, psedoscalars and pseudovectors in four dimensions then
become vector components and tensor components in five dimensions. If
there is a fundamental flaw in this reasoning, I'd like to know exactly
where it is.
John Reed
Theory used an E frame which was the algebra of 4x4 Dirac matrices and had
two time dimensions. He couldn't get his theory to work. Here's a quote
from Kilmister's book "Eddington's Search for a Fundamental Theory":
"The derived E5 does play a physical role in the conventional theory but it
is not that of an extra dimension. Eddington's thinking on E5 was
different. At various times he felt the need, either to have a
five-dimensional theory to give E5 an equal footing with the rest, or to
identify E4, E5 with alternative time-scales in macorscopic and microscopic
systems, or to go into more complex interpretations. Each of these attempts
was in due course abandoned as unworkable."
John Reed
DRLunsford
> 1) Given equation (1), is it fair to think of the gamma^u as being just
> as fundamental to the structure of spacetime as the g^uv, and perhaps
> even more so because the gamma^u have certain features (such as their
> being able to accommodate Dirac spinors which the g_uv alone cannot do,
> and the axial gamma^5 matrix) which are not at all apparent just looking
> at g^uv?
Certainly, they form a basis at each spacetime point. However in
addition to a simple coordinate transformation and its effect on the
tensor g_mn, one can also now consider local rotations of the frame
represented by the 4 gamma_mu.
> 2) If the Dirac gammas can be thought of as the "structure matrices of
> spacetime," then can we also think of the axial gamma^5 as a fifth
> structure matrix of spacetime?
It is a property of Clifford algebras that the unit pseudoscalar on
even dimensional ones anticommutes with all the 2n basis vectors, and
so can be adjoined to the original basis to set up a Clifford algebra
with 2n+1 basis vectors. The unit pseudscalar on this new one will now
be proportional to the identity matrix.
In some ways it is more natural to write the Dirac equation as
(gamma_mu D_mu - M gamma_5) psi = 0
One can now think of M as coming from a fifth coordinate when D_5
operates on psi.
> 4) With U=5, does it make sense to conclude that the existence of the
> gamma^5 is indicative of a fifth spacetime dimension?
Yes, that is a possible interpretation.
> 5) Wherever the gamma^U are taken to be the Dirac matrices
> incorporating pairs of Pauli matrices, the (Minkowskian) metric defined
> by (6) then has diag (g_UV) = (1,-1,-1,-1,1). This gives this "fifth"
> dimension a timelike signature. Does it make any sense, therefore, to
> think of this fifth dimension originating in gamma^5 as a second, "axial
> time" dimension?
Depending on whether or not you add a factor of i when defining the
fifth basis vector, one can get either a timelike or spacelike
signature for it. However the unit pseudoscalar itself in the original
basis is unambiguously defined - it is
I = sqrt(det(g)) eps_1234 gamma_1...gamma_4
The first factor is necessary in order that the permutation symbol eps
be converted into a proper tensor. Since det(g) is negative, the square
root is i and so
I = i gamma_1...gamma_4
and this is the usual definition of gamma_5 in the literature (up to a
minus sign coming from the order 1234 rather than 0123). Since it is
hermitian it corresponds to a timelike extra dimension. Multiplying by
i makes it anti-hermitian and then it would correspond to a spacelike
extra dimension.
You see here the deep connection between i and the unit pseudoscalar in
Clifford algebra.
-drl
mark...@yahoo.com
[Standard recipe for associating a Clifford algebra with an inner
product space deleted]
> 1) Given equation (1), is it fair to think of the gamma^u as being just
> as fundamental to the structure of spacetime as the g^uv, and perhaps
> even more so because the gamma^u have certain features (such as their
> being able to accommodate Dirac spinors which the g_uv alone cannot do,
> and the axial gamma^5 matrix) which are not at all apparent just looking
> at g^uv?
The Clifford algebra comprising the Dirac matrices (gamma0, ...,
gamma3), when viewed as a real-valued linear algebra is equivalent to
M_2(H), the 2x2 matrix algebra of quaternions. It also has numerous
other isomorphisms, which I won't recount here.
When viewed as a complex linear algebra, the extra gamma5 comes into
play. Then it becomes equivalent to M_4(C), the algebra of 4x4 complex
matrices.
The real-valued Clifford algebra that produces M_4(C) is associated
with a 5-dimensional inner product space. There are several inner
products that may yield this algebra. One is (+,+,+,+,-), via the
generators (G_i = gamma_i gamma_5).
Denoting the generators briefly by (i5), this produces the identities:
(i5) (j5) + (j5) (i5) = i5j5 + j5i5 = -ij55 - ji55 =
-2g_{ij} 55 = -2g_{ij},
yielding the signature opposite that of the gamma_i's (+,+,+,-). The
5th generator is just gamma_5, itself, which has the identities
(5)(5) = 1; (i5)5 + 5(i5) = i55 + 5i5 = (i5+5i)5 =
0.
As I outlined in "The Wigner Classification for
Galilei/Poincare/Euclid", this also provides the Clifford algebra
associated with the unifying generalization the Galilei, Poincare and
(4-D) Euclidean groups. (Look under
http://federation.g3z.com/Physics/Index.htm this will soon be converted
to PDF, along with everything else, if you don't have access to Word).
One can even write out an analogue Dirac equation for Galilei, using
this. It reduces equivalently to the Schroedinger equation for
Non-Relativistic Quantum Mechanics.
Igor Khavkine
> When I asked "I am looking for any flaws you can identify in this line
> of thought," I was hoping for a serious response.
I'm sorry that you thought that my response was not serious. I assure
you that it was.
> Yes, the gamma^5 are staring us in the face and someone naiive could
> just say, "gee, that ought to be a fifth spacetime dimension" without
> more than superficial analysis, and I would then agree with the comments
> made. However, let me elaborate my questions, please, because the
> mathematics hangs together quite well and to not seriously consider this
> and give a dismissive answer to me reflects a prejudice in thinking.
What does it mean for the mathematics to "hang together quite well"?
Yes, you've found an algebraic property of the algebra of Dirac gamma
matrices. It is an interesting property and it holds quite generally.
If C' is the complex Clifford algebra constructed over an n-dimentional
(n being an odd number) complex vector space and C is the Clifford
algebra constructed over an (n-1)-dimensional complex vector space.
Then C' is isomorphic to a direct sum of two copies of C. This means
that in any faithful matrix representation of C', we can find a basis
in wich every element of C' is represented by a matrix in block
diagonal form. Each of the blocks gives a matrix representation of C.
See for example:
http://en.wikipedia.org/wiki/Classification_of_Clifford_algebras
This decomposition implies that there exist (several) homomorphisms
(linear multiplication-preserving maps) from C' to C. If P: C' -> C is
such a homomorphism, then {P(e_i),P(e_j)} = P{e_i,e_j} = delta_i,j, as
P(1) = 1, where the e_i are the rank-1 generators of C'. You've found
one such homomorphism from C' to C in the case n=5.
> In other words, can we think of the Dirac gammas as
> the "structure matrices of spacetime" which, via (1), give us an
> alternative way to define a classical spacetime metric?
By construction, the generators of the Clifford algebra correspond to
an orthonormal basis in every tangent space. Knowing what "orthogonal"
means in every tangent space is equivalent to knowing the metric
tensor. So, yes, you can reconstruct the metric tensor from what's
called a Clifford bundle, just like you can from something called the
orthonormal frame bundle.
> Can we equally think, that spacetime is alternatively
> defined by its gamma matrices gamma^u, from which the g^uv may in turn
> be deduced by (1)?
Now, here's the blind step: a space-time is not just the metric. A
space-time is a manifold with a defined on it. A manifold has a fixed
dimension. If the dimension is 4, try as you might, you'll never find 5
linearly independent vectors in a tangent space that are mutually
orthogonal, no matter how you construct the metric tensor.
> 10) Does this lead, at least roughly, to a "many-fingered" time sort of
> notion which I recall Feynman once entertained? What is the
> modern"conventional wisdom" and what other viewpoints are there on such
> things as having more than one timelike dimension, e.g., two timelike
> dimensions, including references which address this point? Has anyone
> ever examined what quantum field theory would look like with a second
> time dimension, that is, has anyone ever explored d = D + 2 QFT, for D=3
> and otherwise? If so, where might I find such examination?
The question of wether theories with more than one time dimension have
been studied is completely separate from all the other questions about
gamma matrices. Yes, such theories have been considered, but apparently
no-one takes them seriously. This question has come up in this group
before. Here's what John Baez had to say on the topic:
news:b45r31$li7$1...@glue.ucr.edu
http://groups.google.ca/group/sci.physics.research/msg/c09890a5f78129ac
Igor
DRLunsford
> The Clifford algebra comprising the Dirac matrices (gamma0, ...,
> gamma3), when viewed as a real-valued linear algebra is equivalent to
> M_2(H), the 2x2 matrix algebra of quaternions. It also has numerous
> other isomorphisms, which I won't recount here.
This isn't the whole story. There is a real representation (Majorana).
> When viewed as a complex linear algebra, the extra gamma5 comes into
> play. Then it becomes equivalent to M_4(C), the algebra of 4x4 complex
> matrices.
This is certainly not true. CL(3,1) and CL(1,3) are, respectively,
H(2) and R(4).
-drl
Jay R. Yablon
news:1158127437....@p79g2000cwp.googlegroups.com...
>
> By construction, the generators of the Clifford algebra correspond to
> an orthonormal basis in every tangent space. Knowing what "orthogonal"
> means in every tangent space is equivalent to knowing the metric
> tensor. So, yes, you can reconstruct the metric tensor from what's
> called a Clifford bundle, just like you can from something called the
> orthonormal frame bundle.
>
>> Can we equally think, that spacetime is alternatively
>> defined by its gamma matrices gamma^u, from which the g^uv may in
>> turn
>> be deduced by (1)?
>
> Now, here's the blind step: a space-time is not just the metric. A
> space-time is a manifold with a defined on it. A manifold has a fixed
> dimension. If the dimension is 4, try as you might, you'll never find
> 5
> linearly independent vectors in a tangent space that are mutually
> orthogonal, no matter how you construct the metric tensor.
Dear Igor:
I always find it easier to think about these things with a concrete
example in mind.
At the link below is a 1.5 page pdf file which lays out a particular
mathematical relationship between a Clifford Algebra based on all five
Dirac gamma matrices, and the equation:
p^u p_u - m^2 = 0
in four spacetime dimensions, with p^u being a contravariant
four-momentum and m being a rest mass.
http://home.nycap.rr.com/jry/Papers/5D%20Nutshell.pdf
I'd be interested in your thoughts on this relationship in the context
of the above discussion, or in any other context that may apply.
Thanks,
Jay.
_____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
Website: http://home.nycap.rr.com/jry/
Jay R. Yablon
>
> (gamma_mu D_mu - M gamma_5) psi = 0
>
> One can now think of M as coming from a fifth coordinate when D_5
> operates on psi.
Hi Danny Ross,
Please take a look at my later post in this thread at:
http://home.nycap.rr.com/jry/Papers/5D%20Nutshell.pdf
I am curious what your view is of this, as this also attempts to
associate the gamma^5 with a mass dimension. In fact, that is my main
reason for interest in the gamma^5 in relation to the dimensionality of
the spacetime manifold.
Jay.
PS: My server does not always receive SPR posts, and this is one of
those that was not forwarded to my server, so I did not see it until
more recently or would have replied sooner.
DRLunsford
> Please take a look at my later post in this thread at:
>
> http://home.nycap.rr.com/jry/Papers/5D%20Nutshell.pdf
? Your calculation is just an identity.
You might want to view the following papers by Dirac:
"A remarkable representation of the O(3,2) de Sitter group" [J. Math.
Phys. 4, 901 (1963)]
"The Electron Wave Equation in De-Sitter Space" [Annals of Mathematics,
2nd Ser., Vol. 36, No. 3 (Jul., 1935), pp. 657-669]
-drl
mark...@yahoo.com
> The Clifford algebra comprising the Dirac matrices (gamma0, ...,
> gamma3), when viewed as a real-valued linear algebra is equivalent to
> M_2(H), the 2x2 matrix algebra of quaternions. It also has numerous
> other isomorphisms, which I won't recount here.
DRLunsford wrote:
> This isn't the whole story. There is a real representation (Majorana).
.. which, however, is not relevant here.
> The Clifford algebra ... when viewed as a REAL-valued linear algebra
(Emphasis mine)
You're referring to the *complex*-ification of the algebra above, which
is an entirely different object...
>> When viewed as a complex linear algebra, the extra gamma5 comes into
>> play. Then it becomes equivalent to M_4(C), the algebra of 4x4 complex
>> matrices.
> This is certainly [sic] not true. CL(3,1) and CL(1,3) are, respectively,
> H(2) and R(4).
.. which is also completely off (as well as redundantly reiterating
what had just been said).
>> when viewed as a REAL-valued linear algebra is equivalent to M_2(H)
i.e. your "H(2)"
>> When viewed as a COMPLEX linear algebra, the extra gamma5 comes into
>> play. Then it becomes equivalent to M_4(C)
i.e. C x M2(H) = M2(C x H) = M4(C)
This is the one that the attribute "has real representations of" refers
to.
You got both items above backwards, because you didn't distinguish
between a real algebra and its complexification. These are two entirely
different objects.
gamma5 does *not* live in the real Clifford algebra, at all; it's i
gamma 5 that does. The addition of gamma5 is tantameount to the
extension of M2(H), by complexification to M4(C).
DRLunsford
> mark...@yahoo.com wrote:
> > The Clifford algebra comprising the Dirac matrices (gamma0, ...,
> > gamma3), when viewed as a real-valued linear algebra is equivalent to
> > M_2(H), the 2x2 matrix algebra of quaternions. It also has numerous
> > other isomorphisms, which I won't recount here.
>
> DRLunsford wrote:
> > This isn't the whole story. There is a real representation (Majorana).
>
> .. which, however, is not relevant here.
Certainly it is. The issue is the difference between Cl(1,3) and
Cl(3,1). In the former case the metric is (+---), the definition of y5
is
y5 = 1/4! eps_mnrs ym yn yr ys = sqrt(det(g)) y0 y1 y2 y3 = i y0 y1 y2
y3
If the ym are all imaginary then so is y5. If the ym are all real, then
y5 is still imaginary. The "i" originates in the square root of the
determinant of g, *and only there*. It is y5, the unit pseudoscalar,
that distinguishes the two metrics.
iy5 does arise as a "complexifier" in this sense. If I make a change of
basis
ym -> ym' = S ym S-1 with S = 1/root2 (1 - iy5)
then in fact
ym' = -iy5 ym
Under this operation, vectors and pseudovectors are interchanged, but
the even subalgebra is left alone. Thus -iy5 mediates a kind of duality
between vectors and pseudovectors.
-drl
mark...@yahoo.com
> > mark...@yahoo.com wrote:
> > > The Clifford algebra comprising the Dirac matrices (gamma0, ...,
> > > M_2(H), the 2x2 matrix algebra of quaternions. It also has numerous
> > > other isomorphisms, which I won't recount here.
(Emphasis mine)
> > DRLunsford wrote:
> > > This isn't the whole story. There is a real representation (Majorana).
> > .. which, however, is not relevant here.
> Certainly it is.
The full sentence was: it's not relevant there, but further down below
-- with the complex(-ified) Dirac algebra. The above was about the REAL
Dirac algebra. One does not distinguish "real" representations of real
linear algebra -- that designation is normally in referenec to complex
algebra. (Yes, a real algebra may also be a complex algebra, but that's
not the ponit here, and the particular algebra C(1,3,0) is not C x
(anything)).
> The issue is the difference between Cl(1,3) and
> Cl(3,1).
.. which has little or nothing to do with what we're talking about
here. You simply misread what you're replying to (as pointed out).
It might help (speaking specifically to Yablon, continuing the first
reply), to get a clearer view of matters to go through the
preliminaries and classification here.
The Clifford algebras, in the most general setting, is an extension of
a vector space V over a field, F, in which one poses identities of the
form a^2 = Q(a), where Q: V -> F is a quadratic form (i.e., Q is
homogeneous of degree 2. so that Q(ka) = k^2 Q(a) comes out, for
coefficients k in F).
Generally, one restricts focus to bi-homogeneous a*a = Q(a,a) and
further restricts focus to those forms that are also linear. But the
more general definitions apply perfectly intact.
For fields F of characteristic other than 2, one can pose the
equivalence
a*a = Q(a,a) <--> ab + ba = 2 Q(a,b)
when Q is symmetric and bi-linear. (Otherwise, one has to revert to the
more general definition Q(a,a) = a*a).
As far as determining the real Clifford algebras goes (the ones over
R), the bi-linear forms over R are characterized by their signature
(+^m, -^n, 0^p), yielding the Clifford algebra C(m,n,p). The ones with
m+n > 1 decompose as follows
C(m+2,n,p) = M2 C(n,m,p); C(m+1,n+1,p) = M2 C(m,n,p); C(m,n+2,p) = H
x C(n,m,p)
where M2 is the (real) 2x2 matrix algebra, and H the quaternions. The
low-lying members of the spectrum are
C(1,0,p) = Lambda(p) + Lambda(p); C(0,1,0) = C = complex
numbers;
the algebra C(0,1,p) for p > 0 don't have a name, as far as I'm aware.
Lambda(p) is the (real) Grassmann algebra for differential forms for a
p-dimensional space; Lambda(0) = R.
It follows from this that C x H = M2(C), H x H = M4. (noting that M2 x
M2 = M4, M2 x M4 = M8, etc). The special case C(1,0,0) = R + R is also
worth noting.
The algebras for m+n+p = 5 can then be worked out. For instance,
C(5,0,0) = M2 C(0,3,0) = M2 x H x C(1,0,0) = M2 x H x (R+R) = M2(H) +
M2(H) = D + D, where D is the (real) Dirac algebra. C(4,1,0) = M2
C(1,2,0) = M2(H) C(0,1,0) = M2(H x C) = M2(M2(C)) = M4(C).
C(4,0,1) = M2 C(0,2,1) = M2(H) x C(0,0,1) = D x Lambda(1).
For the complex algebra, the bi-linear forms are given by their
signature (+^n, 0^p), leading toe the complex Clifford algebras C(n,p),
and the decompositions C(n+2,p) = M2(C) x C(n,p); C(1,p) = Lambda(C;p)
+ Lambda(C;p); C(0,p) = Lambda(C;p); where Lambda(C;p) is the (complex)
Grassmann algebra.
For n = 4, one gets C(3,1,0) = M2 C(1,1,0) = M4; C(1,3,0) = H x
C(1,1,0) = H x M2 = M2(H) = D.
Yablon's question is what higher-dimensional algebra corresponds to the
extension of C(1,3,0) (the Dirac algebra). The addition of the gamma_5
matrices complexifies the Dirac algebra, resulting in C x D = M2(H x C)
= M4(C). These correspond to 5-dimensional real Clifford algebras.
Yablon listed one, another was listed above.
It should be of interest to note that the bilinear form
P^2 - 2MH + a H^2
characterizes the (generalized) mass shell for
Galilean/Euclidean/Poincare' symmetry groups, across the board in all
sectors (Poincare' Luxon/Tachyon/Tardion where a = 1/c^2; Galilei
Tardion/Synchron where a = 0; Euclid Tardion where a < 0). The terms H,
M are repepctively the kinetic energy and "relativistic" mass. No
meaningful E ("total" energy) can be defined in the envelope of the 3
groups, due to the absence of a Galilean limit.
(For tardions P^2 -2MH + a H^2 = 0, the other sectors, however, require
this to be non-zero).
It's natural then to try and find what this translates into, as far as
a Dirac algebra goes, by posing the square root identity:
(a.P + d.M + e.H)^2 = P^2 - 2MH + a H^2;
leading to
a^i a^j + a^j a^i = 2 delta^{ij}; a^i d + d a^i = 0 = a^i e
+ e a^i; i,j = 1,2,3
d^2 = 0; de + ed = -2; e^2 = a.
In fact ... in all cases, the algebra is just C(4,1,0). This includes
the Galilean sectors! One can actually write down a "Dirac" equation
for the Galilean sectors which -- for Tardions -- reduces to the
Schroedinger equation for free particles.