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Quantum Gravity
---------------
Our current understanding of gravity is based on Einstein's
theory of General Relativity that describes gravity as a
curvature within space-time that is influenced by mass.
Conversely, the electromagnetic, weak and strong forces are
described by quantum theories that depend on the emdedding of
particle fields in the flat space-time of special relativity.
Unfortunately, difficulties arise when it comes to constructing
a quantum field theory for gravity with the graviton as the
force particle. This comes about because the theory obtained
is not renormalizable and cannot be used to make meaningful
predictions. Let us see why this is true. We start with the
Einstein-Hilbert action given by:
S = (c^{4}/16πG)∫d^{4}x √(-|g|)R
Where G is Newton's constant and R is the curvature scalar.
Now G can be written quantum mechanically in terms of the
Planck mass, m_{p} as:
G = hc/m_{p}^{2}
If we set h = c = 1 we get:
G = 1/m_{p}^{2}
and (neglecting the factor of 16π):
L = m_{p}^{2}√(-|g|)R
The action becomes:
S = m_{p}^{2}∫d^{4} √(-|g|)R
= (1/G)∫d^{4} √(-|g|)R
Consider small perturbations around flat Minkowski space:
g_{μν} = η_{μν} + Gh_{μν}
We can regard η_{μν} as the 'basic' field, h_{μν} as the theorized
GRAVITON field and G as the coupling constant. The graviton
is expected to be massless (because the gravitational force
appears to have unlimited range like the photon) and must be
a spin-2 boson. A spin 2 particle is also known as a tensor
boson, compared to a spin 0 scalar boson and a spin 1 vector
boson. The spin 2 follows from the fact that the source of
gravitation is the stress-energy tensor which is of second
order (compared to electromagnetism's spin-1 photon, the
source of which is a 4 vector, a first order tensor). Also,
it can be shown that any massless spin 2 field would give
rise to a force indistinguishable from gravitation, because a
massless spin 2 field would couple to the stress–energy tensor
in the same way that gravitational interactions do.
Taylor series review:
f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^{2}/2 + ...
If we set (x - a) = ε so that x = a + ε we get:
f(a + ε) = f(a) + f'(a)ε
Where,
f'(a) = ∂f(x)/∂x|_{x = a}
Now we can do the same for the perturbed metric, g = η + Gh.
We get:
g(η + Gh) = g(η) + Gh∂g(η)/∂g(η) + G^{2}h^{2}∂^{2}g(η)/∂g(η)^{2})/2 + ...
= η + Gh + G^{2}h^{2}/2 + G^{3}h^{3}/6 + ...
Now, schematically,
R ~ ∂Γ/∂x^{μ} + Γ^{2} where Γ ~ ∂g/∂x^{μ} ≡ ∂_{μ}g (Christoffel symbols)
Therefore,
R ~ ∂_{μ}^{2}g + (∂_{μ}g)^{2}
Now (dropping constants for clarity and simplicity),
∂_{μ}g = ∂_{μ}h + Gh∂_{μ}h + G^{2}h^{2}∂_{μ}h + ...
∂_{μ}^{2}g = ∂_{μ}^{2}h + G^{2}h∂_{μ}^{2}h
Therefore,
R ~ ∂_{μ}^{2}h + Gh∂_{μ}^{2}h + (∂_{μ}h + Gh∂_{μ}h + G^{2}h^{2}∂_{μ}h)^{2}
~ ∂_{μ}^{2}h + Gh∂_{μ}^{2}h + (∂_{μ}h + Gh∂_{μ}h)^{2}
~ ∂_{μ}^{2}h + Gh∂_{μ}^{2}h + (∂_{μ}h)^{2} + Gh(∂_{μ}h)^{2} + G^{2}h^{2}(∂_{μ}h)^{2} + Gh(∂_{μ}h)^{2}
+ G^{2}h^{2}(∂_{μ}h)^{2} + G^{3}h^{3}(∂_{μ}h)^{2} + G^{2}h^{2}(∂_{μ}h)^{2} + G^{3}h^{3}(∂_{μ}h)^{2} + G^{4}h^{4}(∂_{μ}h)^{2}
~ (∂_{μ}h)^{2} + Gh(∂_{μ}h)^{2} + G^{2}h^{2}(∂_{μ}h)^{2} + G^{3}h^{3}(∂_{μ}h)^{2} + G^{4}h^{4}(∂_{μ}h)^{2} + ...
Dimensional Comparison
----------------------
Lets now take a look at the dimensions of the Einstein-Hilbert
action under unperturbed and perturbed conditions:
g and h are dimensionless since ds^{2} = g_{μν}dx^{μ}dx^{ν}. However, they
are both functions of x^{μ} so:
[∂_{μ}g] = [∂_{μ}h] = [1/L] = [M]
[d^{4}x] = [L^{4}] = [1/M^{4}]
[Γ] ~ [∂g/∂x^{μ}] ≡ [∂_{μ}g] = [1/L] = [M]
[R] ~ [∂Γ/∂x^{μ} + Γ^{2}] ≡ [∂_{μ}^{2}g + (∂_{μ}g)^{2}] = [1/L^{2}] = [M^{2}]
[G] = [1/L^{2}] = [M^{2}]
In the unperturbed case:
S = (1/G)∫d^{4}x √(-|g|)R
= [m^{2}][m^{-4}][m^{2}]
We see that S is dimensionless (as it needs to be for the Path
Integral formulation in QFT theory).
In the perturbed case:
S = (1/G)∫d^{4}x √(-|g|){(∂h)^{2} + Gh(∂h)^{2} + G^{2}h^{2}(∂h)^{2} + ...}
Graphically, this expansion is an expansion in numbers of
loops in Feynman diagrams. At each loop the whole expression
should be dimensionless. The first term in {} has [m^{2}] so
that is fine. However, the second term has [m^{4}]. In order
to make it equal to [m^{2}], we would need to add 2 more powers
of momentum in the denominator. Because 1/G is of the order
of 10^{16} Kg, the expression becomes more and more divergent the
higher you go in the perturtabive expansion In order to
cancel these divergences it would be necessary to introduce
an infinite number of counterterms which, in terms of
renormalization, makes no sense. The conclusion is that,
unlike other quantum field theories s where the coupling
constant is dimensionless, the gravitational theory is not
renormalizable (at least at very high energies).
Effective Field Theory
----------------------
The problem of non-renormalizability of quantum gravity does
not necessarily mean that quantum mechanics is incompatible
with gravity, only that quantum gravity should be treated as
an EFFECTIVE FIELD THEORY for energies well below the Planck
scale of 10^{19} GeV. An effective field theory appromimates
an underlying physical theory that describes physical phenomena
occurring at a chosen length or energy scale, while ignoring
phenomena at shorter distances. Intuitively, it averages over
the behavior of the underlying theory at shorter length scales
to derive what is hoped to be a simplified model at longer
length scales. Effective field theories typically work best
when there is a large separation between the length scale of
interest and the length scale of the underlying dynamics.