markwh04@yahoo.com
May7-06, 05:00 AM
Starbles@Earthlink.net wrote:
> How is the chromoelectroweak force dealt with in loop quantum gravity?
In order to answer the question, you first have to remove a
preconception: it's not merely "loop quantum gravity", but "loop gauge
theory". The loop representation is generic fo ANY theory whose
canonical variable is a connection.
Even this is not the whole story. The actual underlying approach -- in
its full generality -- should read more like: "path-based
representation of gauge theory", since you can generalize everything
involving loops to general paths. The reason people generally don't is
that one can define in a natural way group structures on loops, whereas
with paths one is forced to resort to the more general concept of a
groupoid -- which some people have an irrational allergy to.
So, there already is a loop (and path) representation for gauge theory
and, in particular, any Yang-Mills theory, including that for SU(3) x
SU(2) x U(1).
For a combined theory, Yang-Mills-Einstein, you would take the product
of the gauge groups for GR and for whatever Yang-Mills field you wanted
to incorporate and work out the loop representation of that.
Equivalently, that amounts to doing loop quantum gravity over the
principle bundle that geometrically represents the Yang-Mills-Einstein
field.
For Yang-Mills-Higgs theories, the corresponding space is a homogeneous
space, which generalises the principle bundle construction. If you
allow the gauge group metric to have a variable metric, this will also
incorporate dilatons. There, too, you can then pull this into the loop
representation.
In fact, the loop representation is universal and can be formulated
INDEPENDENTLY of the gauge theory. This is, in effect, a "universal
pull-back" of all gauge theories. Loop representations of any gauge
theory are then cranked out from the universal pull-back in a
quasi-systematic manner.
There may even be a way to generalize the loop approach to
supersymmetric Yang-Mills theories, where fermion degrees of freedom
are added to the mix. That's something that's never been looked at, as
far as I'm aware. Going further still, it might even be possible to
work one's way all the way to "loopizing" string theory, itself; giving
a loop representation for string theory.
The main application of such a construct would be to resolve the issue
of defining string theory on arbitrary curved backgrounds.
In general, what the loop representation does is map what is known as
the path groupoid of the base manifold to the gauge groupoid of the
gauge group. This is what provides the representation of the connection
associated with the gauge group. There is a nice and simple way of
defining the structure of principle bundles in terms of 2 algebraic
operations (a product and "inner quotient"). The gauge groupoid is
defined in terms of what would then be called the "outer quotient".
That's described in some detail below.
> How is spacetime curvature dealt with in string theory and its
> successors?
That's an open issue, as far as I'm aware, since the general precept
has been a flat spacetime background.
> Do you think it may be necessary to merge the two methods into one to
> obtain a clearer view of physics in general?
>
> I hope to get some answers. :P
Already you can get a glimpse of the features that might be present in
such an amalgamation of a "loop representation of string theory".
Each closed string has, associated with it, a holonomy; and each open
string an element of what is called the gauge groupoid associated with
the gauge.
--------------
Principle Bundles, Gauge Groupoids, and Quotients.
As mentioned, there is a nice and simple way to wrap up all the basic
structure associated with these seemingly complex ideas. If you regard
spacetime as a manifold M, then the extra structure of a gauge group G
gives you an amalgamation P which consists of each point m of M
replaced by a copy of the group G. So, if p is one such element of the
copy associated with a spacetime point m, then corresponding to p will
be the equivalence class
pG = { pg: g in G } <--> m.
Therefore, it's natural just to regard P as the primary geometry and
define M as the quotient
M = P/G = { pG: p in P }.
Each point p of pG = m is then essentially the point m, itself, with a
gauge setting. The point pg is then the point m with the gauge changed
by g. One requires the properties that:
pe = p, where e is the
group identity
(pg)h = p(gh), where gh is
the group product of g and h.
Second, for each point p of P and another point q in the same
equivalence class, pG, one may define the inner quotient p\q as the
gauge setting required to convert p to q. Thus, one has the properties:
p (p\q) = q; p\(pg) = g, where pG
= qG.
Other properties follow from this:
p\p = e; (p\q)(q\r) = p\r; (p\q)^{-1} =
q\p; (pg)\(qh) = g^{-1} p\q h.
A gauge, S, local to a spacetime region U then assigns each point u of
U to one of the members of its coset; so that
S(u)G = u, for each u in
U.
The requirement is that S also be differentiable.
A principle bundle is then defined as the above structure, where it is
assumed that the entire space M can be covered by a family of local
gauges such that for any two gauges S, S' in the family, S(u)\S'(u) is
differentiable, for any point u in the overlap of their corresponding
regions. This inner quotient defines the gauge transformation in the
overlap region from S to S'.
One can define the action of derivatives for the product operation.
Thus, for a constant g and function p(t); or for a constant p and
function g(t), one defines p'(t) g = (p(t) g)'; and (p g(t))' = p
g'(t). It then follows that (p(t) g(t))' = p'(t) g(t) + p(t) g'(t). A
similar property holds for the group G, itself; e.g. (gh)' = g'h + gh'.
For the inner quotient, however, there is no natural way to define the
derivative. What's missing is ability to define how the copy of the
group pG associated with a point m links up with the copy of a group
p'G associated with a nearby point m'.
That's the "connection". A connection then gives you definitions for
p'(t)\q(t); and p(t)\q'(t), wherever p(t)G = q(t)G. The defining
properties would be:
(p'\p) = -p\p'; p\(q' g) = (p\q') g; p\(q
g') = (p\q) g'.
The differential p\dp is what one normally identifies as the
"connection one form". For a given gauge S, the differential B(m) =
S(m)\dS(m) gives you a connection as a function of M, itself, and one
can write down a transformation rule between this and another
connection B1 defined in terms of S1(m) = S(m) x(m) as:
B1 = (Sx)\(dS x + S dx) = x^{-1} (S\dS x + S\S dx) =
x^{-1} B x + x^{-1} dx.
Since the connection tells you how to relate the gauge of one point in
the spacetime M with another nearby point, then along each path, one
may telescope this relation to a relation between the gauges associated
with the endpoints.
A path p(t) in P proceeds along the natural grain of the connection if
p(t)\p'(t) = 0. The corresponding tangent (p\p') is then called
"horizontal".
If m(t) is the path, then picking a point p0 that lies in the class p0G
= m(0), one can define the "lift" of the path m, M(t) as the solution
to the differential equation:
M(0) = p0; dM/dt = S(m) S'(m)*(dm/dt)\M +
S'(m)*(dm/dt) S(m)\M.
The corresponding differential is dM = S dS\M + dS S\M. One can prove
that this is independent of the gauge, by substituting S1 = Sx in the
above and working it out.
The lift is readily proven to be horizontal, M\(dM/dt) = 0, since
M\dM = M\(S dS\M + dS S\M) =
M\S dS\M + M\dS S\M.
But
M\S dS\M = M\S dS\S S\M = -M\S S\dS S\M = -M\dS S\M.
Thus M\dM = 0. Therefore, the lift M is called the horizontal lift of
m. Explicitly, this may be written out as M = m_p0; the horizontal lift
that starts out at point p0.
Knowing what paths in P are horizontal lifts of paths in M is the same
as knowing the connection. A key property is that m_p\m_q = p\q =
constant. Two horizontal lifts have a constant gauge setting relative
to one another.
So, with that background, I can make more explicit what the
correspondence is in the "path" representation of a connection.
A groupoid is a group where the operations are typed; the types of the
form (T->T'). The rules for type-checking are:
I_T: T -> T; where I_T is the "identity"
associated with type T
if g: V->U, h: U->T, then gh: V->T
if g: T->U, then g^{-1}: U->T.
In addition, are the group identities, subject to the type
restrictions:
if g: T -> U, then g I_T = g = I_U g.
if g: W ->V, h: U -> V, k: T -> U, then (gh)k
= g(hk)
if g: T -> U then g g^{-1} = I_U, g^{-1} g =
I_T.
The groupoid concept captures the essence of "reversible
transformation" or "reversible processes".
The paths of a spacetime provide a primary example. The "types" are the
points of M and the groupoid members the paths connecting points. So,
if K is a path connecting point m to m', one would write K: m -> m'.
The inverse K^{-1} is K oriented in the opposite direction. The product
of paths K: m'' -> m', L: m' -> m is KL: m'' -> m, defined as the path
K appended to the path L.
In order to make the rules for identity hold, one requires that ANY
path of the form K K^{-1}, where K: m' -> m, be regarded as equal to
I_m. So, I_m is actually an equivalence class of all paths that start
at m, reach a point m' and then backtrace. More generally, one
therefore also requires that any path be treated as equivalent to
another path that has such a retracing injected somewhere in its midst.
Thus, K L L^{-1} M = K M, subject to the type restrictions.
The gauge groupoid essentially involves what may be called the "outer
quotient". This has the properties:
(p/q) (q/r) = p/r; (p/q) r = p (q\r), if qG =
rG; (p(q\r))/s = p/(s(r\q)).
It also follows, for instance, that p/(qg) = (p g^{-1})/q.
This defines the gauge groupoid, whose types are the points in M, with
p/q: qG -> pG.
It is a function mapping pG to qG such that p/q(q) = p; and p/q(qg) =
pg.
Each path m(t): m -> m' in M has associated with it a "horizontal
lift", M_p(t) which starts out at point p in pG = m and ends up at a
point p' in p'G = m'. This defines the corresponding element of the
gauge groupoid: M_p <-> p'/p. For the point pg, one has M_{pg}(t) =
M_p(t) g, so that the corresponding point is M_{pg} <-> (p'g)/(pg) =
(p'g g^{-1})/p = p'/p.
Thus, a connection may also be defined as a mapping from the path
groupoid to the gauge groupoid.
> How is the chromoelectroweak force dealt with in loop quantum gravity?
In order to answer the question, you first have to remove a
preconception: it's not merely "loop quantum gravity", but "loop gauge
theory". The loop representation is generic fo ANY theory whose
canonical variable is a connection.
Even this is not the whole story. The actual underlying approach -- in
its full generality -- should read more like: "path-based
representation of gauge theory", since you can generalize everything
involving loops to general paths. The reason people generally don't is
that one can define in a natural way group structures on loops, whereas
with paths one is forced to resort to the more general concept of a
groupoid -- which some people have an irrational allergy to.
So, there already is a loop (and path) representation for gauge theory
and, in particular, any Yang-Mills theory, including that for SU(3) x
SU(2) x U(1).
For a combined theory, Yang-Mills-Einstein, you would take the product
of the gauge groups for GR and for whatever Yang-Mills field you wanted
to incorporate and work out the loop representation of that.
Equivalently, that amounts to doing loop quantum gravity over the
principle bundle that geometrically represents the Yang-Mills-Einstein
field.
For Yang-Mills-Higgs theories, the corresponding space is a homogeneous
space, which generalises the principle bundle construction. If you
allow the gauge group metric to have a variable metric, this will also
incorporate dilatons. There, too, you can then pull this into the loop
representation.
In fact, the loop representation is universal and can be formulated
INDEPENDENTLY of the gauge theory. This is, in effect, a "universal
pull-back" of all gauge theories. Loop representations of any gauge
theory are then cranked out from the universal pull-back in a
quasi-systematic manner.
There may even be a way to generalize the loop approach to
supersymmetric Yang-Mills theories, where fermion degrees of freedom
are added to the mix. That's something that's never been looked at, as
far as I'm aware. Going further still, it might even be possible to
work one's way all the way to "loopizing" string theory, itself; giving
a loop representation for string theory.
The main application of such a construct would be to resolve the issue
of defining string theory on arbitrary curved backgrounds.
In general, what the loop representation does is map what is known as
the path groupoid of the base manifold to the gauge groupoid of the
gauge group. This is what provides the representation of the connection
associated with the gauge group. There is a nice and simple way of
defining the structure of principle bundles in terms of 2 algebraic
operations (a product and "inner quotient"). The gauge groupoid is
defined in terms of what would then be called the "outer quotient".
That's described in some detail below.
> How is spacetime curvature dealt with in string theory and its
> successors?
That's an open issue, as far as I'm aware, since the general precept
has been a flat spacetime background.
> Do you think it may be necessary to merge the two methods into one to
> obtain a clearer view of physics in general?
>
> I hope to get some answers. :P
Already you can get a glimpse of the features that might be present in
such an amalgamation of a "loop representation of string theory".
Each closed string has, associated with it, a holonomy; and each open
string an element of what is called the gauge groupoid associated with
the gauge.
--------------
Principle Bundles, Gauge Groupoids, and Quotients.
As mentioned, there is a nice and simple way to wrap up all the basic
structure associated with these seemingly complex ideas. If you regard
spacetime as a manifold M, then the extra structure of a gauge group G
gives you an amalgamation P which consists of each point m of M
replaced by a copy of the group G. So, if p is one such element of the
copy associated with a spacetime point m, then corresponding to p will
be the equivalence class
pG = { pg: g in G } <--> m.
Therefore, it's natural just to regard P as the primary geometry and
define M as the quotient
M = P/G = { pG: p in P }.
Each point p of pG = m is then essentially the point m, itself, with a
gauge setting. The point pg is then the point m with the gauge changed
by g. One requires the properties that:
pe = p, where e is the
group identity
(pg)h = p(gh), where gh is
the group product of g and h.
Second, for each point p of P and another point q in the same
equivalence class, pG, one may define the inner quotient p\q as the
gauge setting required to convert p to q. Thus, one has the properties:
p (p\q) = q; p\(pg) = g, where pG
= qG.
Other properties follow from this:
p\p = e; (p\q)(q\r) = p\r; (p\q)^{-1} =
q\p; (pg)\(qh) = g^{-1} p\q h.
A gauge, S, local to a spacetime region U then assigns each point u of
U to one of the members of its coset; so that
S(u)G = u, for each u in
U.
The requirement is that S also be differentiable.
A principle bundle is then defined as the above structure, where it is
assumed that the entire space M can be covered by a family of local
gauges such that for any two gauges S, S' in the family, S(u)\S'(u) is
differentiable, for any point u in the overlap of their corresponding
regions. This inner quotient defines the gauge transformation in the
overlap region from S to S'.
One can define the action of derivatives for the product operation.
Thus, for a constant g and function p(t); or for a constant p and
function g(t), one defines p'(t) g = (p(t) g)'; and (p g(t))' = p
g'(t). It then follows that (p(t) g(t))' = p'(t) g(t) + p(t) g'(t). A
similar property holds for the group G, itself; e.g. (gh)' = g'h + gh'.
For the inner quotient, however, there is no natural way to define the
derivative. What's missing is ability to define how the copy of the
group pG associated with a point m links up with the copy of a group
p'G associated with a nearby point m'.
That's the "connection". A connection then gives you definitions for
p'(t)\q(t); and p(t)\q'(t), wherever p(t)G = q(t)G. The defining
properties would be:
(p'\p) = -p\p'; p\(q' g) = (p\q') g; p\(q
g') = (p\q) g'.
The differential p\dp is what one normally identifies as the
"connection one form". For a given gauge S, the differential B(m) =
S(m)\dS(m) gives you a connection as a function of M, itself, and one
can write down a transformation rule between this and another
connection B1 defined in terms of S1(m) = S(m) x(m) as:
B1 = (Sx)\(dS x + S dx) = x^{-1} (S\dS x + S\S dx) =
x^{-1} B x + x^{-1} dx.
Since the connection tells you how to relate the gauge of one point in
the spacetime M with another nearby point, then along each path, one
may telescope this relation to a relation between the gauges associated
with the endpoints.
A path p(t) in P proceeds along the natural grain of the connection if
p(t)\p'(t) = 0. The corresponding tangent (p\p') is then called
"horizontal".
If m(t) is the path, then picking a point p0 that lies in the class p0G
= m(0), one can define the "lift" of the path m, M(t) as the solution
to the differential equation:
M(0) = p0; dM/dt = S(m) S'(m)*(dm/dt)\M +
S'(m)*(dm/dt) S(m)\M.
The corresponding differential is dM = S dS\M + dS S\M. One can prove
that this is independent of the gauge, by substituting S1 = Sx in the
above and working it out.
The lift is readily proven to be horizontal, M\(dM/dt) = 0, since
M\dM = M\(S dS\M + dS S\M) =
M\S dS\M + M\dS S\M.
But
M\S dS\M = M\S dS\S S\M = -M\S S\dS S\M = -M\dS S\M.
Thus M\dM = 0. Therefore, the lift M is called the horizontal lift of
m. Explicitly, this may be written out as M = m_p0; the horizontal lift
that starts out at point p0.
Knowing what paths in P are horizontal lifts of paths in M is the same
as knowing the connection. A key property is that m_p\m_q = p\q =
constant. Two horizontal lifts have a constant gauge setting relative
to one another.
So, with that background, I can make more explicit what the
correspondence is in the "path" representation of a connection.
A groupoid is a group where the operations are typed; the types of the
form (T->T'). The rules for type-checking are:
I_T: T -> T; where I_T is the "identity"
associated with type T
if g: V->U, h: U->T, then gh: V->T
if g: T->U, then g^{-1}: U->T.
In addition, are the group identities, subject to the type
restrictions:
if g: T -> U, then g I_T = g = I_U g.
if g: W ->V, h: U -> V, k: T -> U, then (gh)k
= g(hk)
if g: T -> U then g g^{-1} = I_U, g^{-1} g =
I_T.
The groupoid concept captures the essence of "reversible
transformation" or "reversible processes".
The paths of a spacetime provide a primary example. The "types" are the
points of M and the groupoid members the paths connecting points. So,
if K is a path connecting point m to m', one would write K: m -> m'.
The inverse K^{-1} is K oriented in the opposite direction. The product
of paths K: m'' -> m', L: m' -> m is KL: m'' -> m, defined as the path
K appended to the path L.
In order to make the rules for identity hold, one requires that ANY
path of the form K K^{-1}, where K: m' -> m, be regarded as equal to
I_m. So, I_m is actually an equivalence class of all paths that start
at m, reach a point m' and then backtrace. More generally, one
therefore also requires that any path be treated as equivalent to
another path that has such a retracing injected somewhere in its midst.
Thus, K L L^{-1} M = K M, subject to the type restrictions.
The gauge groupoid essentially involves what may be called the "outer
quotient". This has the properties:
(p/q) (q/r) = p/r; (p/q) r = p (q\r), if qG =
rG; (p(q\r))/s = p/(s(r\q)).
It also follows, for instance, that p/(qg) = (p g^{-1})/q.
This defines the gauge groupoid, whose types are the points in M, with
p/q: qG -> pG.
It is a function mapping pG to qG such that p/q(q) = p; and p/q(qg) =
pg.
Each path m(t): m -> m' in M has associated with it a "horizontal
lift", M_p(t) which starts out at point p in pG = m and ends up at a
point p' in p'G = m'. This defines the corresponding element of the
gauge groupoid: M_p <-> p'/p. For the point pg, one has M_{pg}(t) =
M_p(t) g, so that the corresponding point is M_{pg} <-> (p'g)/(pg) =
(p'g g^{-1})/p = p'/p.
Thus, a connection may also be defined as a mapping from the path
groupoid to the gauge groupoid.