14 research outputs found
Quantum Blobs
Quantum blobs are the smallest phase space units of phase space compatible
with the uncertainty principle of quantum mechanics and having the symplectic
group as group of symmetries. Quantum blobs are in a bijective correspondence
with the squeezed coherent states from standard quantum mechanics, of which
they are a phase space picture. This allows us to propose a substitute for
phase space in quantum mechanics. We study the relationship between quantum
blobs with a certain class of level sets defined by Fermi for the purpose of
representing geometrically quantum states.Comment: Prepublication. Dedicated to Basil Hile
Clifford Algebras in Symplectic Geometry and Quantum Mechanics
The necessary appearance of Clifford algebras in the quantum description of
fermions has prompted us to re-examine the fundamental role played by the
quaternion Clifford algebra, C(0,2). This algebra is essentially the geometric
algebra describing the rotational properties of space. Hidden within this
algebra are symplectic structures with Heisenberg algebras at their core. This
algebra also enables us to define a Poisson algebra of all homogeneous
quadratic polynomials on a two-dimensional sub-space, Fa of the Euclidean
three-space. This enables us to construct a Poisson Clifford algebra, H(F), of
a finite dimensional phase space which will carry the dynamics. The quantum
dynamics appears as a realization of H(F) in terms of a Clifford algebra
consisting of Hermitian operators.Comment: 17 page
Causal categories: relativistically interacting processes
A symmetric monoidal category naturally arises as the mathematical structure
that organizes physical systems, processes, and composition thereof, both
sequentially and in parallel. This structure admits a purely graphical
calculus. This paper is concerned with the encoding of a fixed causal structure
within a symmetric monoidal category: causal dependencies will correspond to
topological connectedness in the graphical language. We show that correlations,
either classical or quantum, force terminality of the tensor unit. We also show
that well-definedness of the concept of a global state forces the monoidal
product to be only partially defined, which in turn results in a relativistic
covariance theorem. Except for these assumptions, at no stage do we assume
anything more than purely compositional symmetric-monoidal categorical
structure. We cast these two structural results in terms of a mathematical
entity, which we call a `causal category'. We provide methods of constructing
causal categories, and we study the consequences of these methods for the
general framework of categorical quantum mechanics.Comment: 43 pages, lots of figure
Delayed-Choice Experiments and the Metaphysics of Entanglement
Delayed-choice experiments in quantum mechanics are often taken to undermine a realistic interpretation of the quantum state. More specifically, Healey has recently argued that the phenomenon of delayed-choice entanglement swapping is incompatible with the view that entanglement is a physical relation between quantum systems. This paper argues against these claims. It first reviews two paradigmatic delayed-choice experiments and analyzes their metaphysical implications. It then applies the results of this analysis to the case of entanglement swapping, showing that such experiments pose no threat to realism about entanglement