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UNIMODULAR GRAVITY REDUX
E.A, S. Gonzalez-Martin., M. Herrero-Valea and CP. Martin
JHEP
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Pauli, 1920 (Straumann)
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comment of Henry Whitehead: 'It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial.'
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In Unimodular Gravity the full configuration space is restricted to unimodular metrics
Action principle
Potential energy (not only zero mode) naively decoupled from the gravitational field
How to integrate over unimodular metrics?
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Van der Bij, van Dam and Ng.TDiff has enough freedom to kill the three extra
polarizations: 5-3=2
Transverse Diffeomorphisms
(Generating vector is transverse)
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Convenient new variables
Their variation is unconstrained
Whereas
Hermann_Weyl_ETH-Bib_Portr_00890.jpg (JPEG Image, 46... http://upload.wikimedia.org/wikipedia/commons/7/78/Herma...
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In this variables there is (sort of spurious?) Weyl symmetry in addition to TDiff
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Flat Space
E.A., Blas, Garriga & Verdaguer, NPB
Most general l.c. of dimension 4 operators
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Fierz-Pauli (LDiff)
Two cases of enhanced symmetryBoth propagate spin 2 ONLY
(WTDiff)
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But this is NOT A FIELD REDEFINITION, because it is not invertible
WTDiff is a truncation of Fierz-Pauli obtained by
WTDiff is the linear limit of Unimodular Gravity
Hermann_Weyl_ETH-Bib_Portr_00890.jpg (JPEG Image, 46... http://upload.wikimedia.org/wikipedia/commons/7/78/Herma...
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einstein.jpg (JPEG Image, 800 × 600 pixels) http://calnewport.com/blog/wp-content/uploads/2007/10/einst...
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EM reminiscent of Einstein’s 1919 traceless Theory (E.A. JHEP)
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UG is NOT equivalent to GR in the gauge g=-1
(Although classically it is quite similar)(work in progress)
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The Bianchi identities bring the trace back into the game
The constant piece of the potential, V0, does not source the c.c.
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Quantum corrections
Does the cc get generated by quantum corrections?
Gauge symmetry: Tdiff(M) & Weyl(M)
Need of BRST gauge fixing
Nielsen-Kallosh ghosts (30)
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The operator involving
Is non-minimal.
Usual heat kernel techniques do not work here.
Its determinant has been computed using the (quite unwieldy) Barvinsky-Vilkovisky technique
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Background EM
Topological density modulo a dynamically irrelevant term
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Divergences proportional to the EM can be eliminated by a one-loop field redefinition
Point transformations leave formally invariant the path integral
Counterterms that vanish on-shell are gauge dependent (Kallosh)
Ergo, it is possible to find a gauge where the theory is finite, even off-shell
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Consider a general gauge theory
Gauge fixing & ghost terms are together BRST exact
Let us examine the dependence of the partition function with the gauge fixing
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The classical action is gauge invariant, then BRST invariant
On shell
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The one loop quantum correction reads
This term is non-dynamical; in GR this is not the case (Christensen-Duff)
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This result is not a consequence of
Rather, it is Weyl symmetry that forbids dimension zero operators
It is possible to work out GR in the gauge g=-1. CC is still generated there; there is no symmetry in GR
to prevent it from appearing.
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How certain are we of the belief that the effective string gravitational equations are Diff invariant and not only TDiff
invariant?Impossible to tell the difference from any first quantized
approach...
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Session Two slides
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Background Field Splitting
In order not to clutter formulas
All covariant derivatives etc are defined wrt the background
In terms of a unimodular background
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BRST
TDiff ghosts are transverse
Quadratic piece of the classical action
Non-unimodular background can be easily gotten by going to the Jordan frame
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In order to get a nilpotent BRST operator
It is necessary to define
Better use unconstrained fields
Swapping transversality with gauge symmetry
New ghosts needed in ordet to get
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Batalin-Vilkovisky
(Grassmann number, Ghost number)
Diff
U(1)
Weyl
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Results as above follow
X=(1,-1)
Free invertible kinetic terms
Quadratic in the quantum fields
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The TDiff Sector
Contains the graviton and is akin to the usual Faddeev-Popov gauge fixing
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Completing the square
(This is essentially Faddeev-Popov)
Consider now the fermionic terms
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The knack for choosing
Weyl invariant so that the gauge fixings decouple
Cancel nonmimimal operators for graviton fluctuations
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In terms of unconstrained fields
Choose
Then
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Summarizing
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The Weyl sector
simplest choice
Keeping the free parameter alpha until we put the result on shell yields a powerful check of our computations
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One Loop Effective Action
General one-loop structure
For example
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Minimal operators
The computation of their determinant is a standard Schwinger-DeWitt calculation
Non-minimal operators need some care
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The nonminimal operator
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Barvinsky-Vilkovisky
Schwinger
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In our case the computation simplifies somewhat
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The minimal piece is a standard computation
The nonminimal part is reduced through the Barvinsky-Vilkovisky technique to the computation of
functional traces
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Effective Action= sum of gauge piece plus BRST exact (ghosts)
Christensen & Duff
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On-shell divergences
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Conclusions
Unimodular gravity solves one aspect of the cosmological constant problem, namely, why the vacuum energy does not generate a huge value
for it.
It is a Wilsonian solution to the problem of weighing vacuum energy
(No need to modify UV behavior)
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EM should be derived BEFORE gauge fixing!
In spite of the fact that UG is quite close to GR, which can be worked out in the gauge g=-1, this mechanism does not work in GR because there
is no Weyl symmetry there
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Incidentally, the low energy limit of string theory could as well be UG instead of GR.
With on-shell amplitudes only it is not possible to discriminate the two.
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Backup
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Schwinger- DeWitt
Regularized Effective Action
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Short time expansion
Zeta Function Out of Heat Kernel
Pole Part of the Effective Action
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Laplace-type Operator
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Starting Point: The d’Alembert Operator
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Quartic Operator
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The (b c) System
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The ( , ) System
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The ( ) System
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The ( w ) System
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Functional Traces
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Symmetrized outer products of metric tensors
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