10 research outputs found
Quantum Hamiltonians and Stochastic Jumps
With many Hamiltonians one can naturally associate a |Psi|^2-distributed
Markov process. For nonrelativistic quantum mechanics, this process is in fact
deterministic, and is known as Bohmian mechanics. For the Hamiltonian of a
quantum field theory, it is typically a jump process on the configuration space
of a variable number of particles. We define these processes for regularized
quantum field theories, thereby generalizing previous work of John S. Bell
[Phys. Rep. 137, 49 (1986)] and of ourselves [J. Phys. A: Math. Gen. 36, 4143
(2003)]. We introduce a formula expressing the jump rates in terms of the
interaction Hamiltonian, and establish a condition for finiteness of the rates.Comment: 43 pages LaTeX, no figures. The old version v2 has been divided in
two parts, the first of which is the present version v3, and the second of
which is available as quant-ph/040711
On the Role of Density Matrices in Bohmian Mechanics
It is well known that density matrices can be used in quantum mechanics to
represent the information available to an observer about either a system with a
random wave function (``statistical mixture'') or a system that is entangled
with another system (``reduced density matrix''). We point out another role,
previously unnoticed in the literature, that a density matrix can play: it can
be the ``conditional density matrix,'' conditional on the configuration of the
environment. A precise definition can be given in the context of Bohmian
mechanics, whereas orthodox quantum mechanics is too vague to allow a sharp
definition, except perhaps in special cases. In contrast to statistical and
reduced density matrices, forming the conditional density matrix involves no
averaging. In Bohmian mechanics with spin, the conditional density matrix
replaces the notion of conditional wave function, as the object with the same
dynamical significance as the wave function of a Bohmian system.Comment: 16 pages LaTeX, no figure
Trajectories and Particle Creation and Annihilation in Quantum Field Theory
We develop a theory based on Bohmian mechanics in which particle world lines
can begin and end. Such a theory provides a realist description of creation and
annihilation events and thus a further step towards a "beable-based"
formulation of quantum field theory, as opposed to the usual "observable-based"
formulation which is plagued by the conceptual difficulties--like the
measurement problem--of quantum mechanics.Comment: 11 pages LaTeX, no figures; v2: references added and update
Seven Steps Towards the Classical World
Classical physics is about real objects, like apples falling from trees,
whose motion is governed by Newtonian laws. In standard Quantum Mechanics only
the wave function or the results of measurements exist, and to answer the
question of how the classical world can be part of the quantum world is a
rather formidable task. However, this is not the case for Bohmian mechanics,
which, like classical mechanics, is a theory about real objects. In Bohmian
terms, the problem of the classical limit becomes very simple: when do the
Bohmian trajectories look Newtonian?Comment: 16 pages, LaTeX, uses latexsy
Is the hypothesis about a low entropy initial state of the Universe necessary for explaining the arrow of time?
According to statistical mechanics, microstates of an isolated physical system (say, a gas in a box) at time t0 in a given macrostate of less-than-maximal entropy typically evolve in such a way that the entropy at time t increases with |t-t0| in both time directions. In order to account for the observed entropy increase in only one time direction, the thermodynamic arrow of time, one usually appeals to the hypothesis that the initial state of the Universe was one of very low entropy. In certain recent models of cosmology, however, no hypothesis about the initial state of the Universe is invoked. We discuss how the emergence of a thermodynamic arrow of time in such models can nevertheless be compatible with the above-mentioned consequence of statistical mechanics, appearances to the contrary notwithstanding
Any orthonormal basis in high dimension is uniformly distributed over the sphere
Let Xd be a real or complex Hilbert space of finite but large dimension d, let S(Xd ) denote the unit sphere of Xd, and let u denote the normalized uniform measure on S(Xd ). For a finite subset B of S(Xd ), we may test whether it is approximately uniformly distributed over the sphere by choosing a partition A1, . . . , Am of S(Xd ) and checking whether the fraction of points in B that lie in Ak is close to u(Ak) for each k = 1, . . . , m. We show that if B is any orthonormal basis of Xd and m is not too large, then, if we randomize the test by applying a random rotation to the sets A1, . . . , Am, B will pass the random test with probability close to 1. This statement is related to, but not entailed by, the law of large numbers. An application of this fact in quantum statistical mechanics is briefly described
Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems