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Register a new userQuantum Measurement of Space-Time Events
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Dorje C. Brody Professor
Publication date
2020
fields
Physics
and research's language is
English
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The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive-definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four-momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength.
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We illustrate an isomorphic description of the observable algebra for quantum mechanics in terms of functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product with explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base to essentially translate the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry, hence obtaining the latter from the physical theory itself. We have essentially an extended formalism of the Schrodinger versus Heisenberg picture which we try to describe mathematically as a coordinate map from the phase space, which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry as coordinated by the six position and momentum operators. The observable algebra is taken as an algebra of functions on the latter operators. We advocate the idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of quantum (phase) space. Issues about the kind of noncommutative geometry obtained are also explored.
We study the fractal structure of space-time of two-dimensional quantum gravity coupled to c=-2 conformal matter by means of computer simulations. We find that the intrinsic Hausdorff dimension d_H = 3.58 +/- 0.04. This result supports the conjecture d_H = -2 alpha_1/alpha_{-1}, where alpha_n is the gravitational dressing exponent of a spinless primary field of conformal weight (n+1,n+1), and it disfavours the alternative prediction d_H = 2/|gamma|. On the other hand <l^n> ~ r^{2n} for n>1 with good accuracy, i.e. the boundary length l has an anomalous dimension relative to the area of the surface.
The symmetry of quantum theory under time reversal has long been a subject of controversy because the transition probabilities given by Borns rule do not apply backward in time. Here, we resolve this problem within a rigorous operational probabilistic framework. We argue that reconciling time reversal with the probabilistic rules of the theory requires a notion of operation that permits realizations via both pre- and post-selection. We develop the generalized formulation of quantum theory that stems from this approach and give a precise definition of time-reversal symmetry, emphasizing a previously overlooked distinction between states and effects. We prove an analogue of Wigners theorem, which characterizes all allowed symmetry transformations in this operationally time-symmetric quantum theory. Remarkably, we find larger classes of symmetry transformations than those assumed before. This suggests a possible direction for search of extensions of known physics.
In the understanding of the fundamental interactions, the origin of an arrow of time is viewed as problematic. However, quantum field theory has an arrow of causality, which tells us which time direction is the past lightcone and which is the future. This direction is tied to the conventions used in the quantization procedures. The different possible causal directions have related physics - in this sense they are covariant under time-reversal. However, only one causal direction emerges for a given set of conventions. This causal arrow tells us the direction that scattering reactions proceed. The time direction of scattering in turn tells us the time direction for which entropy increases - the so-called arrow of thermodynamics. This connection is overlooked in most discussions of the arrow of time.
In physics, experiments ultimately inform us as to what constitutes a good theoretical model of any physical concept: physical space should be no exception. The best picture of physical space in Newtonian physics is given by the configuration space of a free particle (or the center of mass of a closed system of particles). This configuration space (as well as phase space), can be constructed as a representation space for the relativity symmetry. From the corresponding quantum symmetry, we illustrate the construction of a quantum configuration space, similar to that of quantum phase space, and recover the classical picture as an approximation through a contraction of the (relativity) symmetry and its representations. The quantum Hilbert space reduces into a sum of one-dimensional representations for the observable algebra, with the only admissible states given by coherent states and position eigenstates for the phase and configuration space pictures, respectively. This analysis, founded firmly on known physics, provides a quantum picture of physical space beyond that of a finite-dimensional manifold, and provides a crucial first link for any theoretical model of quantum spacetime at levels beyond simple quantum mechanics. It also suggests looking at quantum physics from a different perspective.
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