Transcript What can we learn from F = ma?

Quantum Mechanics
on biconformal space
A measurement theory
A gauge theory of classical and quantum mechanics;
hints of quantum gravity
Lara B. Anderson & James T. Wheeler
JTW for MWRM 14
What are the essential elements of a physical
theory?
We will focus on:
• The physical arena
• Dynamical laws
• Measurement theory
Examples: Quantum and Classical Mechanics
QM
Physical
arena
Dynamical
evolution
Measurable
quantities
Phase space (x,p)
Hy = ih ∂y/∂t
<y|y> = V y*y d3x
CM
Euclidean
3-space
F=m
d2x
dt2
<u,v> = u . v
The symmetries of arena, dynamical laws,
and measurement, are often different
Physical arena
Diffeomorphisms
Dynamical laws
Global
Metric/measurement
Local
We may reconcile these differences by
extending all symmetries to agree with that of the
measurement theory.
This is often what gauge theory does.
Gauging
Global
Local
(independent
of position)
(dependent
on position)
We systematically extend to local symmetry with a
connection: a one-form field valued in the Lie algebra
of the symmetry we wish to gauge.
Added to the usual derivative, the connection subtracts
back out the extra terms from the local symmetry.
GR:
∂
∂+G
EM:
∂
∂+A
Gravitational Gauging (Utiyama, Kibble, Isham):
Gravitational gauging differs from other gaugings.
Some symmetry is broken by identifying translational
gauge fields with tangent vectors.
In this way, the gauging specifies the physical
manifold.
Lorentz
∂+w
Poincaré
Translation
e
Local
Lorentz
connection
Translational
gauge field
becomes tetrad
An idea: Let the symmetry of measurement fix
the arena and dynamical laws:
Measurement
Symmetry
Physical arena
Possible
dynamical laws
This makes sense in a gravitational theory: the symmetry
determines the physical manifold, and we were going to
modify (gauge) the dynamical law anyway.
We make three postulates:
1. Measurement:
A. The symmetry is the conformal group
B. Dimensionless scalars are observable
C. We require a spinor representation
• Arena: Determined by biconformal gauging.
3. Dynamical evolution is governed by dilatation
A. Motion is deterministic (Classical Mechanics)
B. Motion is stochastic (Quantum Mechanics)
Postulate 1: Measurement is conformal
We know the symmetry of the world is at least Poincaré.
Also, all measurements are relative to a standard.
The group characterized by these properties is the
conformal group, O(4,2) or its covering group SU(2,2).
Since we know that spinors are needed to describe
fermions, we require SU(2,2).
Notice that the standard of measurement is subject to
the same dynamical evolution as the object of study.
Conformal symmetry
There are fifteen 1-form gauge fields:
•
•
•
•
The vierbein, ea (gauge fields of translations)
The Lorentz spin connection, wab
The co-vierbein fa (special conformal transformations)
The Weyl vector, W (gauge vector of dilatations)
These gauge fields must satisfy the Maurer-Cartan
structure equations, which are just the conformal Lie
algebra in a dual basis.
Consequences of conformal symmetry
Use of the covering group SU(2,2) requires a
complex connection.
We choose generators of Lorentz transformations real.
It follows that:
•
Generators of translations and special conformal
transformations are related by complex
conjugation.
2. The generator of dilatations is imaginary.
N.B. The complex generators still generate real
transformations.
The dilatational gauge vector, W
When we gauge O(4,2), the Weyl vector gives rise to a
positive, real, gauge-dependent factor on transported
lengths:
l = l0 exp  Wadxa
Wa
Wa + af
where f is any real function.
However, comparisons of lengths transported along
different curves may give measurable changes:
l1 / l2 = l01 / l02
exp C-C’ Wadxa
This closed line integral is independent of gauge.
The dilatational gauge vector
When we gauge SU(2,2), the Weyl vector is complex.
This gives a complex factor on transported lengths:
l = l0 exp  Wadxa
Gauge transformations still require real functions f:
Wa
Wa + af
There exists a gauge in which Wa is pure imaginary. In
this gauge, we see that comparisons of lengths now
give measurable phase changes:
l1 / l2 = l01 / l02
exp C-C’ Wadxa
The closed line integral is again independent of gauge.
Postulate 2: The arena for physics
The biconformal gauging of the conformal group
identifies translation and special conformal generators
with the directions of the underlying manifold.
The local Lorentz and dilatational symmetries are as
expected. These give coordinate and scale invariance.
We interpret (ea, fa) as an orthonormal frame field of an
eight dimensional space.
Biconformal space
The solution to the structure equations reveals a
symplectic form
F = ea f a
d (ea fa) = 0
The 8-dim space is therefore a symplectic manifold, with
similar structure to a one particle phase space.
We may also write the symplectic form in coordinates as
F= dxa dya
From this we see that ya is canonically conjugate to xa.
The solution of the structure equations also shows
that Wa is proportional to ya.
Coordinates in biconformal space
Since ya is conjugate to xa, we may think of it as a
generalized momentum.
The geometric units of the eight coordinates support
this,
xa ~ length
ya ~ 1/length
We may introduce any constant with dimensions of
action to write
hya = 2πpa
Postulate 3: Dynamical evolution
We base the dynamical law on the dilatation factor,
l = l0 exp  Wadxa,
considering two alternate versions,
3A. Deterministic evolution
3B. Stochastic evolution
We discuss each in turn.
Postulate 3A: Deterministic evolution
We set the action equal to the integral of the Weyl
vector. The system evolves along paths of extremal
dilatation.
SO(4,2) = -iSSU(2,2)
=  Wadxa = (-2π/h)  padxa
The variation
In varying S, we hold t fixed. In order to preserve the
symplectic bracket
{t, y0} = 1
between x0 = t and y0 we must therefore have
0 = d{t, y0}
= {t, dy0}
= ∂t/∂t ∂(dy0)/ ∂y0
Therefore, the variation dy0 and hence y0, depends
only on the remaining coordinates, xi, t, pi. We set
p0 = H(xi, t, pi)
Vary the action to find the equations of motion
We now vary
SO(4,2) = -iSSU(2,2) =  Wadxa
= (-2π/h)  padxa
= (2π/h)  (Hdt - pidxi)
to find Hamilton’s equations:
dxi /dt = ∂H/∂yi
dyi/dt = -∂H/∂xi
The gauge theory of deterministic biconformal
measurement theory is Hamiltonian mechanics.
The constant h or ih drops out.
No size change occurs.
Postulate 3B: Stochastic evolution
The system evolves probabilistically.
Suppose the probability for a displacement dxa is
inversely proportional to the dilatation along dxa:
P(dxa) ~ 1 / |Wadxa |
For O(4,2), we may say that the ratio of the probabilities
of a system following either of two paths is given by the
ratio of the corresponding dilatation factors:
P( C )/ P( C’ ) = exp  C-C’ Wadxa
Path average
We may ask: What is the probability P(l) of measuring length
l, when the system arrives at the point A? The answer is given
by a path average.
Alternately, ask: Among systems measured to have a fixed
length l, what is the probability that such a system arrives at A?
The answer is the same path average (JTW, 1990):
P(A) =  D[C] exp C Wadxa
Notice that P(A) is not a measurable quantity. It is the
probability of measuring a given magnitude, l, at A. To be
measurable, we must give the probability of finding a
dimensionless ratio, l /l0, at A.
Probability
The probability arriving at A, with a given, fixed
dimensionless ratio, l/l0 is given by the double sum
paths:
P(A) =  D[C,C’] l[C]/l0 [C’]
=
 D[C] D[C’] exp C Wadxa exp -C’ Wadxa
=  D[C] exp C Wadxa  D[C’] exp -C’ Wadxa
= P(A) P*(A)
For O(4,2), these are Wiener (real) path integrals.
For SU(2,2) these are Feynman path integrals.
Quantum Mechanics
The requirement for a standard of measurement
therefore accounts for the use of probability
amplitudes in quantum mechanics
P(A) = P(A) P*(A)
We have arrived at the Feynman path integral
formulation of quantum mechanics. From it, we can
develop the Schrödinger equation, define operators,
and so on.
The postulates also allows derivation of the FokkerPlanck (O(4,2)) or Schrödinger (SU(2,2) equation
directly.
Conclusions
To summarize, we assume:
1. Conformal measurement theory
2. Biconformal gauging of a spinor representation
We find:
3A. Deterministic evolution along extremals of dilatation gives:
• Hamiltonian evolution
• No measurable size change
3B.
Stochastic evolution weighted by dilatation predicts:
• Feynman (not Wiener) path integrals as a result of
the SU(2,2) representation.
• Probability amplitudes as a result of the use of a
standard of measurement.
Where do we go from here?
We now have a geometry which contains both general
relativity (see Wehner & Wheeler, 1999) and a
formulation of quantum physics (see Anderson &
Wheeler, 2004).
It becomes possible to ask questions about the
quantum measurement of curved spaces, i.e., quantum
gravity.
Structure equations
dwab = wcb wac + ea fb - eb fa
dea = eb wab + Wea
dfa = wbafb + faW
dW = 2eb fb
An interesting additional feature is the biconformal
bracket, defined from the imaginary symplectic form:
{xa, yb} =idab
It follows that
{xa, pb} =ihdab
The supersymmetric version of the theory has also been
formulated (Anderson & Wheeler, 2003), and may have
relevance to the Maldacena conjecture.
Example 1: Quantum Mechanics
Physical arena
Phase space (x,p)
Dynamical evolution of y
Hy = ih ∂y/∂t
Correspondence with
measurable numbers
<y|y> = V y*y d3x
Example 2: Newtonian Mechanics
Physical arena
Euclidean 3-space
d 2x
Dynamical evolution of x
F=m
Inner product for
measurable magnitudes
<u,v> = u . v
dt2
Example: Classical mechanics from classical
measurement
Symmetries of Newtonian measurement theory
1. Invariance of the Euclidean line element gives the
Euclidean group (3-dim rotations, plus translations)
2. We actually measure dimensionless ratios of
magnitudes. Invariance of ratios of line elements
gives the conformal group (Euclidean, plus
dilatations and special conformal transformations)
We may use either symmetry.
Classical mechanics from Euclidean measurement
The gauge fields include 3 rotations and 3 translations
These give us the physical arena, and determine a
class of physical theories as follows:
The physical arena:
Three translational gauge fields, ei
= orthonormal frame field on a
3-dim Euclidean manifold
Dynamical laws
Three rotational gauge fields, wij, SO(3) connection,
= local rotational symmetry
We may write any locally SO(3) invariant action.
Classical mechanics from Euclidean measurement
The pair, (ei, wij) is equivalent to the metric and
general coordinate connection, (gij, Gijk). We may
therefore find new dynamical laws using any coordinate
invariant variational principle. For example, let
S =  [gijvivj + f] dt
Variation gives the usual Euler-Lagrange equation in the
form
Dv/dt = ∂f/∂xi
When f = 0, this is the geodesic equation, specifying
Euclidean straight lines. Forces produce deviations from
geodesic motion.
Classical mechanics from conformal
measurement
A similar treatment starting with the 10-dim
conformal group of Euclidean space gives:
1. A 6-dimensional symplectic manifold as the
arena.
2. Local rotational and dilatational symmetry.
3. Hamilton’s equations from a suitable action.
This is a special case of the relativistic version below.
(See also, Wheeler (03), Anderson & Wheeler (04).)
We focus on symmetry, for two reasons
By Noether’s theorem,
symmetry
conservation laws
 Prediction:
Prediction: conserved quantities are
constant
 Interactions:
Gauging extends a symmetry
by introducing new elements into a theory.
These new elements describe interactions.
Conclusions
The gauge theory of Newton’s second law
with respect to the Euclidean group is
Lagrangian mechanics.
The gauge theory of Newton’s second law
with respect to the conformal group is
Hamiltonian mechanics.
We now turn to a more comprehensive, relativistic
treatment of conformal measurement theory.
Tidying up some loose ends…
•The multiparticle case works, even though the space
remains 6 dimensional.
•There is a 6 dimensional metric, but it is consistent
with collisions
ds2 = dx.dx + dx.dy
(Particles must have dx = 0 to collide, regardless of
their relative momenta dy.)
•The extremal value of the integral of the Weyl vector
is zero. Thus, no size change occurs for classical
motion.
There is a suggestion of something deeper…
•Quantum mechanics requires both
position and momentum variables to
make sense.
•Biconformal gauging of Newton’s
theory gives us a space which
automatically has both sets of
variables.
Is it possible that quantum physics takes a
particularly simple form in biconformal space?
We would like to say that the world is really a six (or
eight) dimensional place, in which quantum mechanics
is a natural description of phenomena.
There are some indications that this interpretation of
biconformal space works correctly. In particular, the full
relationship between the inverse-length yi coordinates
and momenta appears to be:
ihyi = 2πpi
The presence of an “i” here turns the dilatational
symmetry into a phase symmetry. If this is true, then the
fundamental symmetry of conformal gauge theory and
the fundamental symmetry of quantum theory coincide.
Conformal gauging of Newton’s law
As it stands, Newton’s second law is invariant under
global rotations, translations and dilatations.
But is not invariant under even global special conformal
transformations.
This is easy to fix: introduce a limited covariant
derivative with a connection specific to global special
conformal transformations.
Conformal gauging of Newton’s law
Now introduce the 10 gauge fields
• Translations give the dreibein, ei
• Special conformal transformations give the codreibein fi
Orthonormal frame field on a 6-dim manifold
3. Rotations give the SO(3) spin connection wij
4. Dilatations give the Weyl vector, W.
Connection for local rotations and dilatations
Conformal gauging of Newton’s law
The gauge fields must satisfy the Maurer-Cartan
structure equations of the conformal Lie algebra.
These are easily solved to reveal a symplectic
form:
d (ek fk) = 0
The units of the six coordinates differ.
Three are correct for position:
Three are correct for momentum:
(xi, length)
(yi, 1/length)
This suggests that the 6-dim space is phase space.
We also find that Wi = -yi
The new dynamical law
Again, we write an action.
Since the geometry is like phase space,
the paths won’t be anything like
geodesics. Path length won’t do.
Instead, we have a new feature - a new
vector field (the Weyl vector) that comes
from the dilatations.
The new dynamical law
Again, write an action. Since we are in a phase space,
geodesics won’t do.
Instead, the conformal geometry that the integral of the
Weyl vector along any path gives the relative physical
size change along that path:
l = l0 exp  (W.v) dt
We take the action to be this integral. Then the
physical paths will be paths of extremal size change.
We’ll add a function just to make it interesting:
S =  [(W.v) + f] dt
Vary the action to find six equations:
Dxi /dt = ∂f/∂yi
Dyi/dt = -∂f/∂xi
If we identify f with the Hamiltonian, these are
Hamilton’s equations.
Note: f occurs naturally in the relativistic version