Dirac Gamma matrices including gamma^5, and the Spacetime Metric g_uv

Jay R. Yablon

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. 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: jyablon@nycap.rr.com

DRLunsford

Jay R. Yablon wrote:

> 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

markwh04@yahoo.com

Jay R. Yablon wrote:
[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

Jay R. Yablon wrote:
> 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/Classif...fford_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.ph...9890a5f78129ac

Igor

DRLunsford

markwh04@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.

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

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:1158127437.726662.39120@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: jyablon@nycap.rr.com
Website: http://home.nycap.rr.com/jry/

Jay R. Yablon

> 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.

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

Jay R. Yablon wrote:

> 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

markwh04@yahoo.com

markwh04@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.

> 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).