Quantum 20/20

This text reviews fundametals and incorporates key themes of quantum physics. One theme contrasts boson condensation and fermion exclusivity. Bose–Einstein condensation is basic to superconductivity, superfluidity and gaseous BEC. Fermion exclusivity leads to compact stars and to atomic structure, and thence to the band structure of metals and semiconductors with applications in material science, modern optics and electronics. A second theme is that a wavefunction at a point, and in particular its phase is unique (ignoring a global phase change). If there are symmetries, conservation laws follow and quantum states which are eigenfunctions of the conserved quantities. By contrast with no particular symmetry topological effects occur such as the Bohm–Aharonov effect: also stable vortex formation in superfluids, superconductors and BEC, all these having quantized circulation of some sort. The quantum Hall effect and quantum spin Hall effect are ab initio topological. A third theme is entanglement: a feature that distinguishes the quantum world from the classical world. This property led Einstein, Podolsky and Rosen to the view that quantum mechanics is an incomplete physical theory. Bell proposed the way that any underlying local hidden variable theory could be, and was experimentally rejected. Powerful tools in quantum optics, including near-term secure communications, rely on entanglement. It was exploited in the the measurement of CP violation in the decay of beauty mesons. A fourth theme is the limitations on measurement precision set by quantum mechanics. These can be circumvented by quantum non-demolition techniques and by squeezing phase space so that the uncertainty is moved to a variable conjugate to that being measured. The boundaries of precision are explored in the measurement of g-2 for the electron, and in the detection of gravitational waves by LIGO; the latter achievement has opened a new window on the Universe. The fifth and last theme is quantum field theory. This is based on local conservation of charges. It reaches its most impressive form in the quantum gauge theories of the strong, electromagnetic and weak interactions, culminating in the discovery of the Higgs. Where particle physics has particles condensed matter has a galaxy of pseudoparticles that exist only in matter and are always in some sense special to particular states of matter. Emergent phenomena in matter are successfully modelled and analysed using quasiparticles and quantum theory. Lessons learned in that way on spontaneous symmetry breaking in superconductivity were the key to constructing a consistent quantum gauge theory of electroweak processes in particle physics.

Particle physics II

Quantum chromodynamics the quantum gauge theory of strong interactions is presented: SU(3) being the (colour) symmetry group. The colour content of strongly interacting particles is described. Gluons, the field particles, carry colour so that they mutually interact – unlike photons. Renormalization leads to the coupling strength declining at large four momentum transfer squared q 2 and to binding of quarks in hadrons at small q 2. The cutoff in the range of the strong interaction is shown to be due to this low q 2 behaviour, despite the gluon being massless. In high energy interactions, say proton-proton collisions, the initial process is a hard (high q 2) parton+parton to parton+parton process. After which the partons undergo softer interactions leading finally to emergent hardrons. Experiments at DESY probing proton structure with electrons are described. An account of electroweak unification completes the book. The weak interaction symmetry group is SUL(2), L specifying handedness. This makes the electroweak symmetry U(1)⊗SUL(2). The weak force carriers, W± and Z0, are massive, which is at odds with the massless carriers required by quantum gauge theories. How the BEH mechanism resolves this problem is described. It involves spontaneous symmetry breaking of the vacuum with scalar fields. The outcome are massive gauge field particles to match the W± and Z0 trio, a massless photon, and a scalar field with a massive particle, the Higgs boson. The experimental programmes that discovered the vector bosons in 1983 and the Higgs in 2012 are described, including features of generic detectors. Finally puzzles revealed by our current understanding are outlined.

The temperature-dependent Yang-Mills trace anomaly as a function of the mass gap

The trace anomaly or, equivalently, the interaction measure is an important thermodynamic quantity/observable, since it is very sensitive to the non-perturbative effects in the gluon plasma. It has been calculated and its analytic and asymptotic properties have been investigated with the combined force of analytic and lattice approaches to the [Formula: see text] Yang-Mills (YM) quantum gauge theory at finite temperature. The first one is based on the effective potential approach for composite operators properly generalized to finite temperature. This makes it possible to introduce into this formalism a dependence on the mass gap [Formula: see text], which is responsible for the large-scale dynamical structure of the QCD ground state. The gluon plasma pressure as a function of the mass gap adjusted by this approach to the corresponding lattice data is shown to be a continuously growing function of temperature [Formula: see text] in the whole temperature range [Formula: see text] with the correct Stefan-Boltzmann limit at very high temperature. The corresponding trace anomaly has a finite jump discontinuity at some characteristic temperature [Formula: see text] with latent heat [Formula: see text]. This is a firm evidence of the first-order phase transition in [Formula: see text] pure gluon plasma. It is exponentially suppressed below [Formula: see text] and has a complicated and rather different dependence on the mass gap and temperature across [Formula: see text]. In the very high temperature limit its non-perturbative part has a power-type fall off.

QED representation for the net of causal loops

The present work tackles the existence of local gauge symmetries in the setting of Algebraic Quantum Field Theory (AQFT). The net of causal loops, previously introduced by the authors, is a model independent construction of a covariant net of local C*-algebras on any 4-dimensional globally hyperbolic space-time, aimed to capture structural properties of any reasonable quantum gauge theory. Representations of this net can be described by causal and covariant connection systems, and local gauge transformations arise as maps between equivalent connection systems. The present paper completes these abstract results, realizing QED as a representation of the net of causal loops in Minkowski space-time. More precisely, we map the quantum electromagnetic field Fμν, not free in general, into a representation of the net of causal loops and show that the corresponding connection system and the local gauge transformations find a counterpart in terms of Fμν.