On some meaningful inner product for real Klein-Gordon fields with positive semi-definite norm

A simple derivation of a meaningful, manifestly covariant inner product for real Klein-Gordon (KG) fields with positive semi-definite norm is provided which turns out - assuming a symmetric bilinear form - to be the real-KG-field limit of the inner product for complex KG fields reviewed by A. Mostafazadeh and F. Zamani in December, 2003, and February, 2006 (quant-ph/0312078, quant-ph/0602151, quant-ph/0602161). It is explicitly shown that the positive semi-definite norm associated with the derived inner product for real KG fields measures the number of active positive and negative energy Fourier modes of the real KG field on the relativistic mass shell. The very existence of an inner product with positive semi-definite norm for the considered real, i.e. neutral, KG fields shows that the metric operator entering the inner product does not contain the charge-conjugation operator. This observation sheds some additional light on the meaning of the C operator in the CPT inner product of PT-symmetric Quantum Mechanics defined by C.M. Bender, D.C. Brody and H.F. Jones.Comment: 8 pages, no figures; correction of obvious typos and marginal improvement of formulation

Kurt Symanzik - a stable fixed point beyond triviality

In 1970 Kurt Symanzik proposed a "precarious" phi**4-theory with a negative quartic coupling constant as a valid candidate for an asymptotically free theory of strong interactions. Symanzik's deep insight in the non-trivial properties of this theory has been overruled since then by the Hermitian intuition of generations of scientists, who considered or consider this actually non-Hermitian highly important theory to be unstable. This short - certainly controversial - communication tries to shed some light on the historical and formalistic context of Symanzik's theory in order to sharpen our (quantum) intuition about non-perturbative theoretical physics between (non)triviality and asymptotic freedom.Comment: 6 pages, no figures, new style files, revised for typos, improved discussion, new references adde

NJL with eight quark interactions: Chiral phases at finite T

The thermodynamic potential and thermal dependence of low lying mass spectra of scalars and pseudoscalars are evaluated in a generalized Nambu -- Jona-Lasinio model, which incorporates eight-quark interactions. These are necessary to stabilize the scalar effective potential for the light and strange quark flavors, which would be otherwise unbounded from below. In addition it turns out that they are also crucial to i) lower the temperature of the chiral transition, in conformity with lattice calculations, ii) sharpen the temperature interval in which the crossover occurs, iii) or even allow for first order transitions to occur with realistic quark mass values, from certain critical values of the parameters. These are unprecedented results which cannot be obtained within the NJL approaches restricted to quartic and six-quark interactions.Comment: 6 pages, 3 figures, Talk presented at SCADRON 70 Workshop on Scalar Mesons and Related Topics, Lisbon, 11-16 February 200

PT-Symmetric Quantum Electrodynamics

The Hamiltonian for quantum electrodynamics becomes non-Hermitian if the unrenormalized electric charge $e$ is taken to be imaginary. However, if one also specifies that the potential $A^\mu$ in such a theory transforms as a pseudovector rather than a vector, then the Hamiltonian becomes PT symmetric. The resulting non-Hermitian theory of electrodynamics is the analog of a spinless quantum field theory in which a pseudoscalar field $\phi$ has a cubic self-interaction of the form $i\phi^3$. The Hamiltonian for this cubic scalar field theory has a positive spectrum, and it has recently been demonstrated that the time evolution of this theory is unitary. The proof of unitarity requires the construction of a new operator called C, which is then used to define an inner product with respect to which the Hamiltonian is self-adjoint. In this paper the corresponding C operator for non-Hermitian quantum electrodynamics is constructed perturbatively. This construction demonstrates the unitarity of the theory. Non-Hermitian quantum electrodynamics is a particularly interesting quantum field theory model because it is asymptotically free.Comment: 9 pages, no figures, revtex

General Aspects of PT-Symmetric and P-Self-Adjoint Quantum Theory in a Krein Space

In our previous work, we proposed a mathematical framework for PT-symmetric quantum theory, and in particular constructed a Krein space in which PT-symmetric operators would naturally act. In this work, we explore and discuss various general consequences and aspects of the theory defined in the Krein space, not only spectral property and PT symmetry breaking but also several issues, crucial for the theory to be physically acceptable, such as time evolution of state vectors, probability interpretation, uncertainty relation, classical-quantum correspondence, completeness, existence of a basis, and so on. In particular, we show that for a given real classical system we can always construct the corresponding PT-symmetric quantum system, which indicates that PT-symmetric theory in the Krein space is another quantization scheme rather than a generalization of the traditional Hermitian one in the Hilbert space. We propose a postulate for an operator to be a physical observable in the framework.Comment: 32 pages, no figures; explanation, discussion and references adde

Impact of excess NOx emissions from diesel cars on air quality, public health and eutrophication in Europe

Diesel cars have been emitting four to seven times more NOx in on-road driving than in type approval tests. These ‘excess emissions’ are a consequence of deliberate design of the vehicle’s after-treatment system, as investigations during the ‘Dieselgate’ scandal have revealed. Here we calculate health and environmental impacts of these excess NOx emissions in all European countries for the year 2013. We use national emissions reported officially under the UNECE Convention for Long-range Transport of Atmospheric Pollutants and employ the EMEP MSC-W Chemistry Transport Model and the GAINS Integrated Assessment Model to determine atmospheric concentrations and resulting impacts. We compare with impacts from hypothetical emissions where light duty diesel vehicles are assumed to emit only as much as their respective type approval limit value or as little as petrol cars of the same age. Excess NO2 concentrations can also have direct health impacts, but these overlap with the impacts from particulate matter (PM) and are not included here. We estimate that almost 10 000 premature deaths from PM2.5 and ozone in the adult population (age >30 years) can be attributed to the NOx emissions from diesel cars and light commercial vehicles in EU28 plus Norway and Switzerland in 2013. About 50% of these could have been avoided if diesel limits had been achieved also in on-road driving; and had diesel cars emitted as little NOx as petrol cars, 80% of these premature deaths could have been avoided. Ecosystem eutrophication impacts (critical load exceedances) from the same diesel vehicles would also have been reduced at similar rates as for the health effects

Images in Christmas Balls

We describe light-reflection properties of spherically curved mirrors, like balls in the Christmas tree. In particular, we study the position of the image which is formed somewhere beyond the surface of a spherical mirror, when an eye observes the image of a pointlike light source. The considered problem, originally posed by Abu Ali Hasan Ibn al-Haitham -- alias Alhazen -- more than a millennium ago, turned out to have the now well known analytic solution of a biquadratic equation, being still of great relevance, e.g. for the aberration-free construction of telescopes. We do not attempt to perform an exhaustive survey of the rich historical and engineering literature on the subject, but develop a simple pedagogical approach to the issue, which we believe to be of continuing interest in view of its maltreating in many high-school textbooks.Comment: 13 pages, 7 figures plain LaTeX; Also see http://cft.fis.uc.pt/eef/mirrors.htm, revised version has simplified formulas, more transparent for a wider audience, one reference adde

Phenomenology of pp->pp eta reaction close to threshold

The recent high statistics measurement of the pp -> pp eta reaction at an excess energy Q=15.5 MeV has been analysed by means of partial wave decomposition of the cross section. Guided by the dominance of the final state 1S0 pp interaction (FSI), we keep only terms involving the FSI enhancement factor. The measured p-p and p-eta effective mass spectra can be well reproduced by lifting the standard on-shell approximation in the enhancement factor and by allowing for a linear energy dependence in the leading 3P0->1S0,s partial wave amplitude. Higher partial waves seem to play only a marginal role