Quantum MechanicsQuantum Mechanicson biconformal spaceon biconformal space

A measurement theory A measurement theory

A gauge theory of classical and quantum A gauge theory of classical and quantum mechanics; hints of quantum gravitymechanics; hints of quantum gravity

Lara B. Anderson & James T. WheelerLara B. Anderson & James T. WheelerJTW for MWRM 14JTW for MWRM 14

What are the essential elements of a What are the essential elements of a physical theory?physical theory?

We will focus on:

• The physical arenaThe physical arena

• Dynamical lawsDynamical laws

• Measurement theoryMeasurement theory

HHih ih ∂∂//∂∂tt

Examples: Examples: QuantumQuantum and Classical MechanicsMechanics

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PhysicalPhysicalarenaarena

DynamicaDynamicallevolution evolution

MeasurabMeasurablelequantitiesquantities

Phase space (x,p)Phase space (x,p)

QMQM CMCM

F = F = mm

dd22xx

dtdt22

<u,v> = u v<u,v> = u v.

EuclideanEuclidean3-space3-space

The The symmetriessymmetries of arena, dynamical of arena, dynamical laws, and measurement, are often laws, and measurement, are often differentdifferent

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.

Physical arena Diffeomorphisms

GaugingGauging

Global Local

(independentof position)

(dependenton position)

We systematically extend to local symmetry with We systematically extend to local symmetry with a a connectionconnection: a one-form field valued in the Lie : a one-form field valued in the Lie algebra of the symmetry we wish to gauge. algebra of the symmetry we wish to gauge.

Added to the usual derivative, the connection Added to the usual derivative, the connection subtracts back out the extra terms from the subtracts back out the extra terms from the local symmetrylocal symmetry.

GR: GR: ∂ ∂ +

EM: EM: ∂ ∂ +

Gravitational Gauging Gravitational Gauging (Utiyama, Kibble, Isham):

Gravitational gauging differs from other gaugings.Gravitational gauging differs from other gaugings.Some symmetry is broken by identifying Some symmetry is broken by identifying translational gauge fields with tangent vectors.translational gauge fields with tangent vectors.

In this way, the gauging specifies the physical In this way, the gauging specifies the physical manifold.manifold.

PoincaréPoincaré

∂ + LorentzLorentz

TranslationTranslation ee

Local Lorentz connection

Translational gauge field becomes tetrad

An idea: Let the symmetry of An idea: Let the symmetry of measurement fix the arena and dynamical measurement fix the arena and dynamical laws:laws:

Possible dynamical laws

Measurement

Symmetry

Physical arena

This makes sense in a gravitational theory: the symmetry determines the physical manifold, and we were going to modify (gauge) the dynamical law anyway.

1.1. Measurement: Measurement: A. The symmetry is the conformal A. The symmetry is the conformal

groupgroupB. Dimensionless scalars are B. Dimensionless scalars are

observableobservableC. We require a spinor representationC. We require a spinor representation

• Arena: Determined by biconformal Arena: Determined by biconformal gauging.gauging.

3.3. Dynamical evolution is governed by Dynamical evolution is governed by dilatationdilatation

A. A. Motion is deterministic (Classical Motion is deterministic (Classical Mechanics)Mechanics)

B. Motion is stochastic (Quantum B. Motion is stochastic (Quantum Mechanics)Mechanics)

We make three postulates:We make three postulates:

Postulate 1: Measurement is Postulate 1: Measurement is conformalconformal

We know the symmetry of the world is at least We know the symmetry of the world is at least Poincaré. Also, all measurements are relative to Poincaré. Also, all measurements are relative to a standard.a standard.

The group characterized by these properties is The group characterized by these properties is the conformal group, O(4,2) or its covering group the conformal group, O(4,2) or its covering group SU(2,2). SU(2,2).

Since we know that spinors are needed to Since we know that spinors are needed to describe fermions, we require SU(2,2).describe fermions, we require SU(2,2).

Notice that the standard of measurement is Notice that the standard of measurement is subject to the same dynamical evolution as the subject to the same dynamical evolution as the object of study.object of study.

There are fifteen 1-form gauge fields:There are fifteen 1-form gauge fields:

• The vierbein, The vierbein, eeaa (gauge fields of translations)(gauge fields of translations)• The Lorentz spin connection, The Lorentz spin connection, aa

bb

• The co-vierbein The co-vierbein ffaa (special conformal (special conformal transformations)transformations)

• The Weyl vector, The Weyl vector, WW (gauge vector of (gauge vector of dilatations)dilatations)

Conformal symmetryConformal symmetry

These gauge fields must satisfy the Maurer-These gauge fields must satisfy the Maurer-Cartan structure equations, which are just the Cartan structure equations, which are just the conformal Lie algebra in a dual basis.conformal Lie algebra in a dual basis.

Consequences of conformal symmetryConsequences of conformal symmetry

Use of the covering group SU(2,2) Use of the covering group SU(2,2) requiresrequires a a complex connection.complex connection.

We choose generators of Lorentz We choose generators of Lorentz transformations real.transformations real.

It follows that:It follows that:

• Generators of translations and special Generators of translations and special conformal transformations are related by conformal transformations are related by complex conjugation.complex conjugation.

2. The generator of dilatations is imaginary.2. The generator of dilatations is imaginary.

N.B. The complex generators still generate N.B. The complex generators still generate realreal transformations.transformations.

When we gauge O(4,2), the Weyl vector gives When we gauge O(4,2), the Weyl vector gives rise to a positive, real, gauge-dependent factor rise to a positive, real, gauge-dependent factor on transported lengths:on transported lengths:

ll = = ll00 exp exp WWdxdx

The dilatational gauge The dilatational gauge vector, Wvector, W

ll11 / / ll22 = = ll0101 / / ll0202 exp exp C-C’C-C’WWdxdx

This closed line integral is independent of gauge.This closed line integral is independent of gauge.

WW WW

where where is any real function.is any real function.

However, comparisons of lengths transported However, comparisons of lengths transported along different curves may give measurable along different curves may give measurable changes:changes:

When we gauge SU(2,2), the Weyl vector is When we gauge SU(2,2), the Weyl vector is complex. This gives a complex factor on complex. This gives a complex factor on transported lengths:transported lengths:ll = = ll00 exp exp WWdxdx

Gauge transformations still require Gauge transformations still require realreal functions functions

The dilatational gauge The dilatational gauge vectorvector

ll11 / / ll22 = = ll0101 / / ll0202 exp exp C-C’C-C’WWdxdx

The closed line integral is again independent of The closed line integral is again independent of gauge.gauge.

WW WW

There exists a gauge in which There exists a gauge in which WW is pure is pure

imaginary. In this gauge, we see that imaginary. In this gauge, we see that comparisons of lengths now give measurable comparisons of lengths now give measurable phasephase changes: changes:

Postulate 2: The arena for physicsPostulate 2: The arena for physics

The biconformal gauging of the conformal group The biconformal gauging of the conformal group identifies translation and special conformal identifies translation and special conformal generators with the directions of the underlying generators with the directions of the underlying manifold.manifold.The local Lorentz and dilatational symmetries The local Lorentz and dilatational symmetries are as expected. These give coordinate and scale are as expected. These give coordinate and scale invariance.invariance.We interpret (We interpret (eeaa, , ffaa) as an orthonormal frame ) as an orthonormal frame

field of anfield of aneight dimensional space.eight dimensional space.

The solution to the structure equations reveals a The solution to the structure equations reveals a symplectic formsymplectic form

eeaa ffaa

d d ((eeaa ffaa) = 0 ) = 0

The 8-dim space is therefore a symplectic The 8-dim space is therefore a symplectic manifold, with similar structure to a one particle manifold, with similar structure to a one particle phase space.phase space.

We may also write the symplectic form in We may also write the symplectic form in coordinates ascoordinates as

ddxx dydy

From this we see that yFrom this we see that y is canonically conjugate is canonically conjugate to xto x..

Biconformal Biconformal spacespace

The solution of the structure equations also The solution of the structure equations also shows that Wshows that W is proportional to y is proportional to y..

Since ySince y is conjugate to x is conjugate to x, we may think of it as , we may think of it as a generalized momentum.a generalized momentum.

The geometric units of the eight coordinates The geometric units of the eight coordinates support this,support this,

xx ~ length ~ lengthyy ~ 1/length ~ 1/length

We may introduce any constant with dimensions We may introduce any constant with dimensions of action to writeof action to write

hyhy= 2πp= 2πp

Coordinates in biconformal Coordinates in biconformal spacespace

Postulate 3: Dynamical Postulate 3: Dynamical evolutionevolution

We base the dynamical law on the dilatation We base the dynamical law on the dilatation factor,factor,

ll = = ll00 exp exp WWdxdx

considering two alternate versions,considering two alternate versions,

3A. Deterministic evolution3A. Deterministic evolution

3B. Stochastic evolution3B. Stochastic evolution

We discuss each in turn.We discuss each in turn.

Postulate 3A: Deterministic evolutionPostulate 3A: Deterministic evolution

We set the action equal to the integral of the We set the action equal to the integral of the WeylWeyl

vector. The system evolves along paths of vector. The system evolves along paths of extremal extremal

dilatation. dilatation. SSO(4,2)O(4,2) = -iS = -iSSU(2,2) SU(2,2) == WWdxdx = (-2π/h)= (-2π/h)

ppdxdx

The variationThe variation

In varying S, we hold t fixed. In order to In varying S, we hold t fixed. In order to preserve the symplectic bracketpreserve the symplectic bracket

{t, y{t, y00} = 1} = 1

between xbetween x00 = t and y = t and y00 we must therefore have we must therefore have

0 = 0 = {t, y{t, y00}} = {t, = {t, yy00}}

= ∂t/∂t ∂(= ∂t/∂t ∂(yy00)/ ∂y)/ ∂y00

Therefore, the variation Therefore, the variation yy00 and hence y and hence y00, , depends only on the remaining coordinates, depends only on the remaining coordinates, xxii, t, p, t, pii. We set. We set

pp00 = H(x = H(xii, t, p, t, pii))

dxdxi i /dt = ∂/dt = ∂/∂y/∂yii dydyii/dt = -∂/dt = -∂/∂x/∂xii

Vary the action to find the equations of Vary the action to find the equations of motionmotion

SSO(4,2)O(4,2) = -iS = -iSSU(2,2) SU(2,2) == WWdxdx

= (-2π/h)= (-2π/h) ppdxdx

= (= (2π/h)2π/h) Hdt - pHdt - piidxdxii))

We now varyWe now vary

to find Hamilton’s equations:to find Hamilton’s equations:

The constant h or ih drops out.The constant h or ih drops out.No size change occurs.No size change occurs.

The gauge theory of deterministic The gauge theory of deterministic biconformal measurement theory is biconformal measurement theory is Hamiltonian mechanics.Hamiltonian mechanics.

Postulate 3B: Stochastic evolutionPostulate 3B: Stochastic evolution

The system evolves probabilistically.The system evolves probabilistically.

Suppose the probability for a displacement dxSuppose the probability for a displacement dx is is

inversely proportional to the dilatation along dxinversely proportional to the dilatation along dx::PP(dx(dx) ~ 1 / ) ~ 1 / ||WWdxdx||

For O(4,2), we may say that the ratio of the For O(4,2), we may say that the ratio of the probabilities of a system following either of two probabilities of a system following either of two paths is given by the ratio of the corresponding paths is given by the ratio of the corresponding dilatation factors:dilatation factors:

PP( C )/ ( C )/ PP( C’ ) = exp ( C’ ) = exp C-C’C-C’WWdxdx

Path averagePath averageWe may ask: What is the probability We may ask: What is the probability PP((ll) of ) of measuring length measuring length ll, when the system arrives at the , when the system arrives at the point A? The answer is given by a path average.point A? The answer is given by a path average.

Alternately, ask: Among systems measured to have a Alternately, ask: Among systems measured to have a fixed length fixed length ll, what is the probability that such a , what is the probability that such a system arrives at A? The answer is the same path system arrives at A? The answer is the same path average (JTW, 1990): average (JTW, 1990):

PP(A) = (A) = D[C] exp D[C] exp CCWWdxdx

Notice that Notice that PP(A) is not a measurable quantity. It is the (A) is not a measurable quantity. It is the probability of measuring a given magnitude, probability of measuring a given magnitude, ll, at A. To , at A. To be measurable, we must give the probability of finding be measurable, we must give the probability of finding a dimensionless ratio, a dimensionless ratio, ll / /ll00, at A., at A.

ProbabilityProbability

The probability arriving at A, with a given, The probability arriving at A, with a given, fixed dimensionless ratio, fixed dimensionless ratio, ll//ll00 is given by the is given by the double sum paths:double sum paths:

P(A) = P(A) = D[C,C’] D[C,C’] ll[C]/[C]/ll0 0 [C’][C’]

= = D[C] D[C’] exp D[C] D[C’] exp CCWWdxdx exp - exp -C’C’WWdxdx

= = D[C] exp D[C] exp CCWWdxdx D[C’] exp - D[C’] exp -C’C’WWdxdx

= = PP(A) (A) PP**(A)(A)For O(4,2), these are For O(4,2), these are WienerWiener (real) path (real) path integrals. For SU(2,2) these are integrals. For SU(2,2) these are FeynmanFeynman path integrals.path integrals.

The requirement for a standard of The requirement for a standard of measurement therefore accounts for the use measurement therefore accounts for the use of probability amplitudes in quantum of probability amplitudes in quantum mechanicsmechanics

P(A) = P(A) = PP(A) (A) PP**(A)(A)

Quantum MechanicsQuantum Mechanics

We have arrived at the Feynman path integral We have arrived at the Feynman path integral formulation of quantum mechanics. From it, formulation of quantum mechanics. From it, we can develop the Schrödinger equation, we can develop the Schrödinger equation, define operators, and so on.define operators, and so on.

The postulates also allows derivation of the The postulates also allows derivation of the Fokker-Planck (O(4,2)) or Schrödinger Fokker-Planck (O(4,2)) or Schrödinger (SU(2,2) equation directly.(SU(2,2) equation directly.

ConclusionsConclusions

To summarize, we assume:To summarize, we assume:1.1. Conformal measurement theoryConformal measurement theory2.2. Biconformal gauging of a spinor representationBiconformal gauging of a spinor representation

We find:We find:3A. Deterministic evolution along extremals of 3A. Deterministic evolution along extremals of

dilatation gives:dilatation gives:• Hamiltonian evolutionHamiltonian evolution• No measurable size changeNo measurable size change

3B.3B. Stochastic evolution weighted by dilatation Stochastic evolution weighted by dilatation predicts:predicts:• Feynman (not Wiener) path integrals as a Feynman (not Wiener) path integrals as a

result of the SU(2,2) representation.result of the SU(2,2) representation.• Probability amplitudes as a result of the use of Probability amplitudes as a result of the use of

a standard of measurement.a standard of measurement.

Where do we go from Where do we go from here?here?We now have a geometry which contains both We now have a geometry which contains both general relativity (see Wehner & Wheeler, 1999) general relativity (see Wehner & Wheeler, 1999) and a formulation of quantum physics (see and a formulation of quantum physics (see Anderson & Wheeler, 2004).Anderson & Wheeler, 2004).

It becomes possible to ask questions about the It becomes possible to ask questions about the quantum measurement of curved spaces, i.e., quantum measurement of curved spaces, i.e., quantum gravity. quantum gravity.

ddaabb = = cc

b b aacc + + eeaa

ffbb - - eeb b ffaa

dedeaa = e = eb b aabb + + WeWeaa

dfdfaa = = bbaaffbb + + ffaaWW

dW = 2edW = 2ebb ffbb

Structure equationsStructure equations

An interesting additional feature is the An interesting additional feature is the biconformal bracket, defined from the biconformal bracket, defined from the imaginaryimaginary symplectic form:symplectic form:

{x{xyy} =i} =i

It follows thatIt follows that{x{xpp} =ih} =ih

The supersymmetric version of the theory has The supersymmetric version of the theory has also been formulated (Anderson & Wheeler, also been formulated (Anderson & Wheeler, 2003), and may have relevance to the Maldacena 2003), and may have relevance to the Maldacena conjecture.conjecture.

HHih ih ∂∂//∂∂tt

Example 1: Example 1: QuantumQuantum MechanicsMechanics

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Physical arenaPhysical arena

Dynamical evolution Dynamical evolution of of

Correspondence with Correspondence with measurable numbersmeasurable numbers

Phase space (x,p)Phase space (x,p)

Example 2: Newtonian Example 2: Newtonian MechanicsMechanics

F = F = mm

dd22xx

dtdt22

<u,v> = u v<u,v> = u v.

Euclidean 3-spaceEuclidean 3-spacePhysical arenaPhysical arena

Dynamical evolution of Dynamical evolution of x x

Inner product for Inner product for measurable measurable magnitudesmagnitudes

Symmetries of Newtonian measurement Symmetries of Newtonian measurement theorytheory

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.

Example: Classical mechanics from classical Example: Classical mechanics from classical measurement measurement

Classical mechanics from Euclidean Classical mechanics from Euclidean measurementmeasurement

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, eei

= orthonormal frame field on a

3-dim Euclidean manifoldDynamical lawsDynamical laws

Three rotational gauge fields, ij, SO(3)

connection, = local rotational symmetry

We may write any locally SO(3) invariant action.

S =S = ggijijvviivvj j + + ] dt] dt

Classical mechanics from Euclidean Classical mechanics from Euclidean measurementmeasurement

Variation gives the usual Euler-Lagrange equation Variation gives the usual Euler-Lagrange equation in the formin the form

Dv/dt = ∂Dv/dt = ∂/∂x/∂xii

When When this is the geodesic equation, this is the geodesic equation, specifying Euclidean straight lines. Forces produce specifying Euclidean straight lines. Forces produce deviations from geodesic motion.deviations from geodesic motion.

The pair, (The pair, (eeii, , iijj) is equivalent to the metric ) is equivalent to the metric

and general coordinate connection, (gand general coordinate connection, (gijij, , ijk). We We

may therefore find new dynamical laws using any may therefore find new dynamical laws using any coordinate invariant variational principle. For coordinate invariant variational principle. For example, letexample, let

Classical mechanics from conformal Classical mechanics from conformal measurementmeasurement

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 This is a special case of the relativistic version

below.below.

(See also, Wheeler (03), Anderson & Wheeler (04).)(See also, Wheeler (03), Anderson & Wheeler (04).)

PredictionPrediction: By Noether’s theorem, : By Noether’s theorem, symmetry symmetry

conservation lawsconservation laws

Prediction: conserved quantities are Prediction: conserved quantities are constantconstant

InteractionsInteractions: Gauging extends a symmetry : Gauging extends a symmetry by introducing new elements into a theory.by introducing new elements into a theory.

These new elements describe interactionsThese new elements describe interactions..

We focus on We focus on symmetrysymmetry, for two , for two reasonsreasons

The gauge theory of Newton’s second The gauge theory of Newton’s second law with respect to the conformal law with respect to the conformal group is Hamiltonian mechanics.group is Hamiltonian mechanics.

The gauge theory of Newton’s second The gauge theory of Newton’s second law with respect to the Euclidean law with respect to the Euclidean group is Lagrangian mechanics.group is Lagrangian mechanics.

ConclusionsConclusions

We now turn to a more comprehensive, We now turn to a more comprehensive, relativistic treatment of conformal measurement relativistic treatment of conformal measurement theory.theory.

Tidying up some loose ends…Tidying up some loose ends…

•The multiparticle case works, even though The multiparticle case works, even though the space remains 6 dimensional.the space remains 6 dimensional.

•There is a 6 dimensional metric, but it is There is a 6 dimensional metric, but it is consistent with collisionsconsistent with collisions

dsds22 = dx = dx..dx + dxdx + dx..dydy(Particles must have dx = 0 to collide, (Particles must have dx = 0 to collide, regardless of their relative momenta dy.)regardless of their relative momenta dy.)

•The extremal value of the integral of the Weyl The extremal value of the integral of the Weyl vector is zero. Thus, no size change occurs for vector is zero. Thus, no size change occurs for classical motion.classical motion.

There is a suggestion of something deeper…There is a suggestion of something deeper…

Is it possible that quantum physics takes a Is it possible that quantum physics takes a particularly simple form in biconformal particularly simple form in biconformal space? space?

•Quantum mechanics Quantum mechanics requiresrequires both position and momentum both position and momentum variables to make sense.variables to make sense.

•Biconformal gauging of Biconformal gauging of Newton’s theory gives us a Newton’s theory gives us a space which automatically has space which automatically has both sets of variables.both sets of variables.

There are some indications that this There are some indications that this interpretation of biconformal space works interpretation of biconformal space works correctly. In particular, the full relationship correctly. In particular, the full relationship between the inverse-length ybetween the inverse-length yii coordinates and coordinates and momenta appears to be:momenta appears to be:

ihyihyii = 2πp = 2πpiiThe presence of an “i” here turns the The presence of an “i” here turns the dilatational symmetry into a phase symmetry. If dilatational symmetry into a phase symmetry. If this is true, then the fundamental symmetry of this is true, then the fundamental symmetry of conformal gauge theory and the fundamental conformal gauge theory and the fundamental symmetry of quantum theory coincide.symmetry of quantum theory coincide.

We would like to say that the world is really a six We would like to say that the world is really a six (or eight) dimensional place, in which quantum (or eight) dimensional place, in which quantum mechanics is a natural description of mechanics is a natural description of phenomena.phenomena.

Conformal gauging of Newton’s Conformal gauging of Newton’s lawlaw

As it stands, Newton’s second law is invariant As it stands, Newton’s second law is invariant under global rotations, translations and under global rotations, translations and dilatations.dilatations.

But is But is notnot invariant under even invariant under even globalglobal special special conformal transformations.conformal transformations.

This is easy to fix: introduce a limited covariant This is easy to fix: introduce a limited covariant derivative with a connection specific to global derivative with a connection specific to global special conformal transformations. special conformal transformations.

Conformal gauging of Newton’s lawConformal gauging of Newton’s law

Now introduce the 10 gauge fieldsNow introduce the 10 gauge fields

• Translations give the dreibein, Translations give the dreibein, eeii

• Special conformal transformations give Special conformal transformations give the co-dreibein the co-dreibein ffii

Orthonormal frame field on a 6-dim Orthonormal frame field on a 6-dim manifoldmanifold

3.3. Rotations give the SO(3) spin connection Rotations give the SO(3) spin connection ii

jj

4.4. Dilatations give the Weyl vector, Dilatations give the Weyl vector, WW. .

Connection for local rotations and Connection for local rotations and dilatationsdilatations

Conformal gauging of Newton’s Conformal gauging of Newton’s lawlaw

The gauge fields must satisfy the Maurer-The gauge fields must satisfy the Maurer-Cartan structure equations of the Cartan structure equations of the conformal Lie algebra.conformal Lie algebra.

These are easily solved to reveal a These are easily solved to reveal a symplectic form:symplectic form:

d d ((eekk ffkk) = 0 ) = 0 The units of the six coordinates differ.The units of the six coordinates differ.

Three are correct for position: (xThree are correct for position: (xii, , length)length) Three are correct for momentum: (yThree are correct for momentum: (yii, , 1/length)1/length)

This suggests that the 6-dim space is phase space.This suggests that the 6-dim space is phase space.

We also find that WWe also find that Wii = -y = -yii

Again, we write an action.Again, we write an action.

Since the geometry is like phase Since the geometry is like phase space, the paths won’t be anything space, the paths won’t be anything like geodesics. Path length won’t like geodesics. Path length won’t do.do.

Instead, we have a new feature - a Instead, we have a new feature - a new vector field (the Weyl vector) new vector field (the Weyl vector) that comes from the dilatations.that comes from the dilatations.

The new dynamical lawThe new dynamical law

We’ll add a function just to make it interesting:We’ll add a function just to make it interesting:

Again, write an action. Since we are in a phase Again, write an action. Since we are in a phase space, geodesics won’t do.space, geodesics won’t do.

Instead, the conformal geometry that the Instead, the conformal geometry that the integral of the Weyl vector along any path gives integral of the Weyl vector along any path gives the relative physical size change along that path: the relative physical size change along that path:

ll = = ll00 exp exp (W(W..v) dtv) dt

S =S = [(W[(W..v) + v) + ] dt] dt

We take the action to be this integral. Then theWe take the action to be this integral. Then thephysical paths will be paths of extremal size change.physical paths will be paths of extremal size change.

The new dynamical lawThe new dynamical law

DxDxi i /dt = ∂/dt = ∂/∂y/∂yii DyDyii/dt = -∂/dt = -∂/∂x/∂xii

If we identify If we identify with the Hamiltonian, these with the Hamiltonian, these are Hamilton’s equations. are Hamilton’s equations.

Vary the action to find six Vary the action to find six equations:equations:

Note: Note: occurs naturally in the relativistic versionoccurs naturally in the relativistic version