Rock Brentwood
Feb10-09, 06:00 AM
In reply to:
John Baez (s.p.r., 2000 September 28):
Subject: Re: Spin foams and gauge theories
http://groups.google.com/group/sci.physics.research/msg/ec0270507139dfe1?dmode=source
>>All of this sounds a lot like a rehashing of SL(2, C) gauge gravity, which
>>has already been known to be renormalizable (in the free field case) since
>>the 1980's.
>"The free field case" - are you talking about some linear field theory
>or what here? Please tell me exactly what the "free field case of SL(2,C)
>gauge gravity" is.
The "free field" case (in what you're replying to) means gravity, by
itself, not coupled to matter -- the Ricci vacuum. In other words, if
all you care about is quantizing the pure gravitational field (the
'exterior solution'), then it suffices to treat gravity simply as a
gauge theory for SL(2,C) on a CLASSICAL Riemannian background.
That is: the metric remains classical, as does the Levi-Civita
connection. But the *native* connection (along with the contorsion and
torsion) are quantized as part of the Lorentz gauge field.
>There are lots of different ways to formulate general relativity as
>a theory involving an SL(2,C) or SO(3,1) connection. Many of
>these involve a tetrad field in addition to the connection. The
>Barrett-Crane model is related to an approach where the fields are
>an SO(3,1) connection A and an so(3,1)-valued 2-form B on spacetime.
All formulations of gravitational theories can be translated between
the different geometric underpinnings used (metric-affine, pure metric
+ torsion, Weitzenboeck Poincare' gauge gravity, etc.) The *relevant*
fact here is that the exercise shows, in a fairly clear way, that the
extra effort in trying to "quantize the geometry" or "discretize the
manifold", itself, is much ado about nothing -- overkill. In fact:
overkill that may even be fatal -- overreaching. I strongly suspect
that somewhere down the line there is a No Go theorem on the non-
existence of "full quantum gravity" (that which quantizes both the
metric and connection) for 4 and more dimensions (where gravity
becomes non-trivial).
A more fruitful approach, in the long run, is to simply put the two
pieces together: the classical Riemannian metric sector and the
quantized connection + quantized fields ... and then, *instead* of
worrying about how to create "hybridized semi-classical"
backpropagation equations, just simply ignore the issue and treat it
as irrelevant. Instead of regarding backpropagation as something that
comes from a hybridized dynamics, the more fruitful approach may be to
try to get it for free by regarding it as a "diffeomorphism anomaly"
on the action integral, itself; along the lines:
delta (integral_W L_M) = integral_W (delta L_M) + O(h)
L_M = matter Lagrangian
with the anomalous term
O(h) = h-bar/(16 pi A) integral_W R root(-g) d^4 x
arising as a product of moving the boundary of the region W (horizon
effects, entanglement entropy). This may be the place where one can
directly insert the Bekenstein bound as a postulate.
In quantized form, denoting the integral on the left by S_M, one may
have a classico-quantum hybrid field theory with the following
characteristics:
(1) the vacuum is HIGHLY degenerate (one vacuum state for *each*
subbundle FM subset of TM, where TM is the tangent bundle of the
manifold M, FM the orthonormal frame bundle)
(2) superselection on different FM -- no coherent superposition
between two states that disagree on what frames are the free fall
frames (each sees the other's vacuum as a *mixed* state, not a pure
quantum state).
(3) a variational of the form, ultimately arising from the above-
mentioned "diffeomorphism anomaly":
delta <FM| E-hat |FM> = (1/(16 pi A) integral R root(-g) d^4 x) E-
hat
of the quantized operator
E-hat = exp(i S_M-hat/h-bar)
where S_M-hat is the quantized action; where both R and g on the right
are the Riemannian curvature scalar and metric associated with the
frame bundle FM; and |FM><FM| is the vacuum state for the sector
corresponding to FM.
John Baez (s.p.r., 2000 September 28):
Subject: Re: Spin foams and gauge theories
http://groups.google.com/group/sci.physics.research/msg/ec0270507139dfe1?dmode=source
>>All of this sounds a lot like a rehashing of SL(2, C) gauge gravity, which
>>has already been known to be renormalizable (in the free field case) since
>>the 1980's.
>"The free field case" - are you talking about some linear field theory
>or what here? Please tell me exactly what the "free field case of SL(2,C)
>gauge gravity" is.
The "free field" case (in what you're replying to) means gravity, by
itself, not coupled to matter -- the Ricci vacuum. In other words, if
all you care about is quantizing the pure gravitational field (the
'exterior solution'), then it suffices to treat gravity simply as a
gauge theory for SL(2,C) on a CLASSICAL Riemannian background.
That is: the metric remains classical, as does the Levi-Civita
connection. But the *native* connection (along with the contorsion and
torsion) are quantized as part of the Lorentz gauge field.
>There are lots of different ways to formulate general relativity as
>a theory involving an SL(2,C) or SO(3,1) connection. Many of
>these involve a tetrad field in addition to the connection. The
>Barrett-Crane model is related to an approach where the fields are
>an SO(3,1) connection A and an so(3,1)-valued 2-form B on spacetime.
All formulations of gravitational theories can be translated between
the different geometric underpinnings used (metric-affine, pure metric
+ torsion, Weitzenboeck Poincare' gauge gravity, etc.) The *relevant*
fact here is that the exercise shows, in a fairly clear way, that the
extra effort in trying to "quantize the geometry" or "discretize the
manifold", itself, is much ado about nothing -- overkill. In fact:
overkill that may even be fatal -- overreaching. I strongly suspect
that somewhere down the line there is a No Go theorem on the non-
existence of "full quantum gravity" (that which quantizes both the
metric and connection) for 4 and more dimensions (where gravity
becomes non-trivial).
A more fruitful approach, in the long run, is to simply put the two
pieces together: the classical Riemannian metric sector and the
quantized connection + quantized fields ... and then, *instead* of
worrying about how to create "hybridized semi-classical"
backpropagation equations, just simply ignore the issue and treat it
as irrelevant. Instead of regarding backpropagation as something that
comes from a hybridized dynamics, the more fruitful approach may be to
try to get it for free by regarding it as a "diffeomorphism anomaly"
on the action integral, itself; along the lines:
delta (integral_W L_M) = integral_W (delta L_M) + O(h)
L_M = matter Lagrangian
with the anomalous term
O(h) = h-bar/(16 pi A) integral_W R root(-g) d^4 x
arising as a product of moving the boundary of the region W (horizon
effects, entanglement entropy). This may be the place where one can
directly insert the Bekenstein bound as a postulate.
In quantized form, denoting the integral on the left by S_M, one may
have a classico-quantum hybrid field theory with the following
characteristics:
(1) the vacuum is HIGHLY degenerate (one vacuum state for *each*
subbundle FM subset of TM, where TM is the tangent bundle of the
manifold M, FM the orthonormal frame bundle)
(2) superselection on different FM -- no coherent superposition
between two states that disagree on what frames are the free fall
frames (each sees the other's vacuum as a *mixed* state, not a pure
quantum state).
(3) a variational of the form, ultimately arising from the above-
mentioned "diffeomorphism anomaly":
delta <FM| E-hat |FM> = (1/(16 pi A) integral R root(-g) d^4 x) E-
hat
of the quantized operator
E-hat = exp(i S_M-hat/h-bar)
where S_M-hat is the quantized action; where both R and g on the right
are the Riemannian curvature scalar and metric associated with the
frame bundle FM; and |FM><FM| is the vacuum state for the sector
corresponding to FM.