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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