Field
Theory Seminar
James
T. Wheeler:
Examples
of dilatational Noether currents
The
symplectic dual
Supersymmetry
Electromagnetism (U(1) gauge theory) extended to biconformal space
The Weyl vector and dilatational curvature
February
19 and 24, March 1, 2010
Lecture: Examples
of dilatational Noether currents; The symplectic dual
Q: Examples of Noether currents of
scale invariant theories
Dilatational
Noether current in Lagrangian
mechanics
The usual form of the Lagrangian for classical mechanics may be treated as scale
invariant if we assign appropriate conformal weights to all dimensionful
constants. For example, the action for the simple harmonic oscillator is scale
invariant if we scale the coordinate and the mass appropriately. The resulting
system only has the trivial solution x = 0 because we havenÕt included a
kinetic term for the mass.
A
more interesting model is one where the action is built from dimensionless
fractions. Notice that the Noether current is the
inner product of the momentum with the inverse position vector.
Dilatational
Noether current in field theory: Weyl
geometry
Field
theory gives more realistic models. This action is the one Dirac used in his
Large Numbers Hypothesis paper. It describes a scalar field coupled to gravity
in a Weyl geometry. Assigning the usual +2 conformal
weight to the metric, it is easy to determine the scaling weights necessary for
the action to be invariant.
(Lower
right) This action actually describes general relativity on a Riemannian
manifold. The easiest way to see this is to define a new metric given by the square of the scalar field
times the old metric. When this scale invariant combination is substituted into
the action, the scalar field decouples. The k4 term becomes a
cosmological constant.
Without
the simplifying substitution, things are more complicated. Varying the action (Palatini variation) The connection variation shows, with a
little algebra (see the trace in the slide f), that the rescaled metric is covariantly constant.
The
k variation gives a simple
wave equation for k,
coupled to the scalar curvature, and we find the conserved current, which is
built from k and its derivatives. The
conservation of the current is consistent with the field equation. Finally, we
note why there is no mass term for k. In a Weyl geometry, dimensionful constants become fields. If we want the mass
to be covariantly constant, then we must satisfy the integrability condition. But the integrability
condition implies pure-gauge Weyl vector, so the
geometry becomes Riemannian. It turns out that this is the case anyway, as seen
by comparing the covariant constancy of the rescaled metric to the
non-vanishing, Weyl-covariant derivative of the
original metric. The comparison shows that the Weyl
vector is pure gauge.
Dilatational
Noether current in field theory: biconformal geometry
Finally,
consider biconformal gravity. The linear curvature gravity action involves only
the dilatational curvature and the Lorentz curvature, both of which are gauge
invariant. Only the solder and co-solder forms scale, and since the action does
not contain their derivatives, there is no conserved current. There are curvature-quadratic terms,
e.g., torsion/co-torsion, which give rise to dilatational Noether
currents.
The symplectic dual
Yang-Mills
theories in biconformal spaces
Yang-Mills theories in general
Starting
from any Lie group, gauging leads to a field strength. We may take this field
strength 2-form, wedge with its Hodge dual, and contract using the Killing
metric (if the group has one). Integrating the resulting n-form yields a
suitable action functional. The potential is used to couple in a
phenomenological source term. Alternatively, introducing the group-covariant
derivative into the field equation for potential sources will give not only the
potential-current coupling in this minimal form, but also determine the form of
the current in terms of fundamental fields.
If
we try to apply the method to gravity, we get extra curvature-quadratic terms
because the Hodge dual depends on the metric.
The symplectic structure of biconformal space
Now
consider biconformal spaces. Because of the doubled dimension (and therefore not in other conformal gauge
theories), the dilatation equation becomes a symplectic form, manifestly exact
and non-degenerate. It remains
symplectic even in curved biconformal spaces.
In
biconformal spaces, the restriction of the Killing metric to the base manifold
is non-degenerate. This fact is also unique to this gauging. We can find a
canonically conjugate, orthogonal set of basis forms, each of which spans a
metric submanifold. When we start the construction
with a Euclidean space, the
induced metric on the configuration space is necessarily Minkowski.
Now
consider a 4-dim Yang-Mills theory, and ask how to extend it to an equivalent
8-dim biconformal theory. There is essentially only one thing we can add which
is linear in the 4-dim fields. We find its field strength. The use of the Hodge
dual complicates the theory considerably.
However,
there is an alternative. In addition to the Hodge dual, we may define a
symplectic dual, in which the role of the metric is played by the symplectic
form. As long as the basis is canonical, the volume element of the space is
also independent of the metric.
The symplectic dual, and a question on sypersymmetry
Lecture: Supersymmetry;
the symplectic dual
First,
a question on supersymmetry
Q: Supersymmetry
The
Coleman-Mandula theorem shows that any candidate Lie
group for at Theory Of Everything (TOE, i.e. electroweak, strong, and gravity)
cannot satisfy reasonable constraints on a quantum field theory unless the
gravity group enters only in a direct product with the internal symmetry of the
Grand Unified Theory (GUT, i.e., unified electroweak and strong). For example,
SU(5) gives a tight unification of SU(3) x SU(2) x U(1) that predicts new
particles because it is not a direct product.
The
new X and Y particles in SU(5) theory, called leptoquarks,
lead to proton decay. The predicted lifetime (~ 1031 years) is
shorted than the observed limits, ruling out the simplest version of the
theory. Supersymmetry groups avoid the Coleman Mandula
theorem by allowing anticommutators in the algebra.
By using such graded Lie algebras
instead of just Lie algebras, we find transformations that can mix fields of
different spin.
As
a simple (approximate) example, consider a scalar field together with a spinor field. A fermionic
transformation variable can mix the two.
Supersymmetry
(SUSY) transformations generally anticommute to give
translation generators.
The
fermionic generators are often written as derivative
operators with respect to anticommuting coordinates
(Question
on SUSY: f some stray
calculations)
The
expansion of a scalar superfield. A scalar function
of normal and anticommuting coordinates has a finite
Taylor series in the anticommuting fermionic coordinates, since each must square to zero.
Carrying out this expansion gives a supersymmetric multiplet, in this case consisting of four scalar fields,
four spinor fields, and a vector field. Some of these
are removable by supersymmetry transformations.
The symplectic dual
Returning
to our previous discussion.
A
careful definition of the Hodge and symplectic duals. Both involve the Levi-Civita tensor, which normally has a factor of the square
root of the determinant of the metric. However, the dual basis of biconformal
space allows the Levi-Civita tensor to be broken into
two submanifolds of opposite tensor and scaling
type. This means that the metric
dependent volume factors cancel in the full volume form. The symplectic dual is therefore
completely independent of the metric.
The
symplectic dual of a 1-form
The
exterior derivative of the dual of a 1-form; some notational conventions.
The
boundary operator (center, right) gives the difference
of two divergences. The exterior derivative of this will give the field part of
the U(1) field equation.
The
symplectic dual remains independent of the metric as long as the basis is
canonical. If we transform one coordinate without changing the other to keep it
canonical, we pick up the usual metric dependence of the volume form.
Symplectic geometry, canonical transformations and the Darboux theorem
Lecture:
Symplectic
geometry, canonical transformations and the Darboux
theorem
A
symplectic form is a closed, nondegenerate, 2-form,
which may play a role in defining a geometry much the way the metric is the
defining field in Riemannian geometry. There is an important difference: by the
Darboux theorem, there always exist coordinates in
which the symplectic form is a constant, unit-antisymmetric
matrix. This means that there are no derivatives, and therefore no analog of
curvature from the symplectic form. It is possible to have both a metric and a
symplectic form, making curved symplectic manifolds possible.
Coordinate
transformations which preserve the form of the symplectic form are called
canonical. We may use the symplectic form to define the Poisson bracket;
canonical transformations preserve Poisson brackets.
We
prove that we can put the symplectic form into standard form at a point. WeÕll
look later at a proof that this form can be achieved throughout a finite neighborhood
of every point.
We
seek a diagonalizing transformation
Some
simple results for matrices. Hermitian matrices may
be diagonalized and have real eigenvalues;
anti-Hermitian matrices may be written as i times an Hermitian
matrix and so can be diagonalized with imaginary eigenvalues. Normal matrices are those that commute with
their adjoints; writing a normal matrix as the sum of
its Hermitian and anti-Hermitian
parts we see that these two parts commute. They are therefore simultaneously
diagonable.
Since
the symplectic form is antisymmetric, it commutes
with its adjoint and is therefore normal. Normal
matrices are diagonable. The eigenvalues
of an antisymmetric matrix will be pure imaginary:
A
simple rotation brings the matrix to standard form
Some
musings on how to show the constancy on a region.
