1. 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

2. What are the essential elements of a physical theory? We will focus on: The physical arena The physical arena Dynamical laws Dynamical laws Measurement theory Measurement theory

3. H  ih ∂  /∂t Examples: QuantumMechanics Examples: Quantum and Classical Mechanics  V  d 3 x <  V  d 3 x PhysicalarenaDynamicalevolutionMeasurablequantities Phase space (x,p) QMCM F = m d2xd2xd2xd2x dt 2 = u v = u v.Euclidean3-space

4. The symmetries of arena, dynamical laws, and measurement, are often different Dynamical lawsGlobal 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

5. 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 Added to the usual derivative, the connection subtracts back out the extra terms from the local symmetry. GR: GR: ∂ ∂ +  EM: EM: ∂ ∂ + 

6. Gravitational Gauging 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. Poincaré ∂ + Lorentz Translation e Local Lorentz connection Translational gauge field becomes tetrad

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

8. 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. Arena: Determined by biconformal gauging. 3. Dynamical evolution is governed by dilatation A. Motion is deterministic (Classical Mechanics) B. Motion is stochastic (Quantum Mechanics) We make three postulates:

9. 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.

10. There are fifteen 1-form gauge fields: The vierbein, e a (gauge fields of translations) The vierbein, e a (gauge fields of translations) The Lorentz spin connection,  a b The Lorentz spin connection,  a b The co-vierbein f a (special conformal transformations) The co-vierbein f a (special conformal transformations) The Weyl vector, W (gauge vector of dilatations) The Weyl vector, W (gauge vector of dilatations) Conformal symmetry These gauge fields must satisfy the Maurer-Cartan structure equations, which are just the conformal Lie algebra in a dual basis.

11. 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. 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.

12. When we gauge O(4,2), the Weyl vector gives rise to a positive, real, gauge-dependent factor on transported lengths: l = l 0 exp  W  dx  The dilatational gauge vector, W l 1 / l 2 = l 01 / l 02 exp  C-C’  W  dx  This closed line integral is independent of gauge. WWWW W     where  is any real function. However, comparisons of lengths transported along different curves may give measurable changes:

13. When we gauge SU(2,2), the Weyl vector is complex. This gives a complex factor on transported lengths: l = l 0 exp  W  dx  Gauge transformations still require real functions  The dilatational gauge vector l 1 / l 2 = l 01 / l 02 exp  C-C’  W  dx  The closed line integral is again independent of gauge. WWWW W     There exists a gauge in which W  is pure imaginary. In this gauge, we see that comparisons of lengths now give measurable phase changes:

14. 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 (e a, f a ) as an orthonormal frame field of an eight dimensional space.

15. The solution to the structure equations reveals a symplectic form  e a f a d (e a f a ) = 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  dx  dy  From this we see that y  is canonically conjugate to x . Biconformal space The solution of the structure equations also shows that W  is proportional to y .

16. Since y  is conjugate to x , we may think of it as a generalized momentum. The geometric units of the eight coordinates support this, x  ~ length y  ~ 1/length We may introduce any constant with dimensions of action to write hy   = 2πp  Coordinates in biconformal space

17. Postulate 3: Dynamical evolution We base the dynamical law on the dilatation factor, l = l 0 exp   W  dx   l = l 0 exp   W  dx   considering two alternate versions, 3A. Deterministic evolution 3B. Stochastic evolution We discuss each in turn.

18. Postulate 3A: Deterministic evolution We set the action equal to the integral of the Weyl vector. The system evolves along paths of extremal dilatation. S O(4,2) = -iS SU(2,2) =  W  dx  = (-2π/h)  p  dx  S O(4,2) = -iS SU(2,2) =  W  dx  = (-2π/h)  p  dx 

19. The variation In varying S, we hold t fixed. In order to preserve the symplectic bracket {t, y 0 } = 1 between x 0 = t and y 0 we must therefore have 0 =  {t, y 0 } = {t,  y 0 } = {t,  y 0 } = ∂t/∂t ∂(  y 0 )/ ∂y 0 = ∂t/∂t ∂(  y 0 )/ ∂y 0 Therefore, the variation  y 0 and hence y 0, depends only on the remaining coordinates, x i, t, p i. We set p 0 = H(x i, t, p i )

20. dx i /dt = ∂  /∂y i dy i /dt = -∂  /∂x i Vary the action to find the equations of motion S O(4,2) = -iS SU(2,2) =  W  dx  = (-2π/h)  p  dx  = (-2π/h)  p  dx  = ( 2π/h)  Hdt - p i dx i ) = ( 2π/h)  Hdt - p i dx i ) We now vary to find Hamilton’s equations: The constant h or ih drops out. No size change occurs. The gauge theory of deterministic biconformal measurement theory is Hamiltonian mechanics.

21. Postulate 3B: Stochastic evolution The system evolves probabilistically. Suppose the probability for a displacement dx  is inversely proportional to the dilatation along dx  : P (dx  ) ~ 1 / | W  dx  | 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’  W  dx 

22. 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  W  dx  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 / l 0, at A.

23. Probability The probability arriving at A, with a given, fixed dimensionless ratio, l / l 0 is given by the double sum paths: P(A) =  D[C,C’] l [C]/ l 0 [C’] P(A) =  D[C,C’] l [C]/ l 0 [C’] =  D[C] D[C’] exp  C  W  dx  exp -  C’  W  dx  =  D[C] D[C’] exp  C  W  dx  exp -  C’  W  dx  =  D[C] exp  C  W  dx   D[C’] exp -  C’  W  dx  =  D[C] exp  C  W  dx   D[C’] exp -  C’  W  dx  = P (A) P * (A) = P (A) P * (A) For O(4,2), these are Wiener (real) path integrals. For SU(2,2) these are Feynman path integrals.

24. The requirement for a standard of measurement therefore accounts for the use of probability amplitudes in quantum mechanics P(A) = P (A) P * (A) Quantum Mechanics 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 Fokker- Planck (O(4,2)) or Schrödinger (SU(2,2) equation directly.

25. 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 Hamiltonian evolution No measurable size change 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. 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. Probability amplitudes as a result of the use of a standard of measurement.

26. 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.

28. d  a b =  c b  a c + e a f b - e b f a de a = e b  a b + We a df a =  b a f b + f a W dW = 2e b f b Structure equations

29. An interesting additional feature is the biconformal bracket, defined from the imaginary symplectic form: {x   y  } =i    It follows that {x   p  } =ih    The supersymmetric version of the theory has also been formulated (Anderson & Wheeler, 2003), and may have relevance to the Maldacena conjecture.

30. H  ih ∂  /∂t H  ih ∂  /∂t Example 1: QuantumMechanics Example 1: Quantum Mechanics  V  d 3 x <  V  d 3 x Physical arena Dynamical evolution of  Correspondence with measurable numbers Phase space (x,p)

31. Example 2: Newtonian Mechanics F = m d2xd2xd2xd2x dt 2 = u v = u v. Euclidean 3-space Physical arena Dynamical evolution of x Inner product for measurable magnitudes

32. Symmetries of Newtonian measurement theory 1. Invariance of the Euclidean line element gives the Euclidean group (3-dim rotations, plus translations) 2. 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 measurement

33. 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: e Three translational gauge fields, e i = orthonormal frame field on a 3-dim Euclidean manifold Dynamical laws  Three rotational gauge fields,  i j, SO(3) connection, = local rotational symmetry We may write any locally SO(3) invariant action.

34. S =  g ij v i v j +  ] dt Classical mechanics from Euclidean measurement Variation gives the usual Euler-Lagrange equation in the form Dv/dt = ∂  /∂x i When  this is the geodesic equation, specifying Euclidean straight lines. Forces produce deviations from geodesic motion. The pair, (e i,  i j ) is equivalent to the metric and general coordinate connection, (g ij,  We may therefore find new dynamical laws using any coordinate invariant variational principle. For example, let The pair, (e i,  i j ) is equivalent to the metric and general coordinate connection, (g ij,  i jk ). We may therefore find new dynamical laws using any coordinate invariant variational principle. For example, let

35. Classical mechanics from conformal measurement A similar treatment starting with the 10-dim conformal group of Euclidean space gives: 1. 1. A 6-dimensional symplectic manifold as the arena. 2. 2. Local rotational and dilatational symmetry. 3. 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).)

36. Prediction: By Noether’s theorem, Prediction: By Noether’s theorem, symmetry conservation laws Prediction: conserved quantities are constant Interactions: Gauging extends a symmetry by introducing new elements into a theory. Interactions: Gauging extends a symmetry by introducing new elements into a theory. These new elements describe interactions. We focus on symmetry, for two reasons

37. The gauge theory of Newton’s second law with respect to the conformal group is Hamiltonian mechanics. The gauge theory of Newton’s second law with respect to the Euclidean group is Lagrangian mechanics. Conclusions We now turn to a more comprehensive, relativistic treatment of conformal measurement theory.

38. Tidying up some loose ends… The multiparticle case works, even though the space remains 6 dimensional.The multiparticle case works, even though the space remains 6 dimensional. There is a 6 dimensional metric, but it is consistent with collisionsThere is a 6 dimensional metric, but it is consistent with collisions ds 2 = 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.The extremal value of the integral of the Weyl vector is zero. Thus, no size change occurs for classical motion.

39. There is a suggestion of something deeper… Is it possible that quantum physics takes a particularly simple form in biconformal space? Quantum mechanics requires both position and momentum variables to make sense.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.Biconformal gauging of Newton’s theory gives us a space which automatically has both sets of variables.

40. There are some indications that this interpretation of biconformal space works correctly. In particular, the full relationship between the inverse-length y i coordinates and momenta appears to be: ihy i = 2πp i 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. 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.

41. 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.

42. Conformal gauging of Newton’s law Now introduce the 10 gauge fields Translations give the dreibein, e i Translations give the dreibein, e i Special conformal transformations give the co- dreibein f i Special conformal transformations give the co- dreibein f i Orthonormal frame field on a 6-dim manifold 3. Rotations give the SO(3) spin connection  i j 4. Dilatations give the Weyl vector, W. Connection for local rotations and dilatations

43. 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 (e k f k ) = 0 The units of the six coordinates differ. Three are correct for position: (x i, length) Three are correct for position: (x i, length) Three are correct for momentum: (y i, 1/length) Three are correct for momentum: (y i, 1/length) This suggests that the 6-dim space is phase space. We also find that W i = -y i

44. 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

45. We’ll add a function just to make it interesting: 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 = l 0 exp  (W. v) dt S =  [(W. v) +  ] dt We take the action to be this integral. Then the physical paths will be paths of extremal size change. The new dynamical law

46. Dx i /dt = ∂  /∂y i Dy i /dt = -∂  /∂x i If we identify  with the Hamiltonian, these are Hamilton’s equations. Vary the action to find six equations: Note:  occurs naturally in the relativistic version