Spectra of phase point operators in odd prime dimensions and the extended Clifford group

We analyse the role of the Extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbons et al for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.Comment: Latex, 19page

Gaps between Jets in the High Energy Limit

We use perturbative QCD to calculate the parton level cross section for the production of two jets that are far apart in rapidity, subject to a limitation on the total transverse momentum Q0 in the interjet region. We specifically address the question of how to combine the approach which sums all leading logarithms in Q/Q0 (where Q is the jet transverse momentum) with the BFKL approach, in which leading logarithms of the scattering energy are summed. This paper constitutes progress towards the simultaneous summation of all important logarithms. Using an "all orders" matching, we are able to obtain results for the cross section which correctly reproduce the two approaches in the appropriate limits.Comment: 29 pages, 9 figures, minor corrections to text and improved figure

Optimization of the e-e- option for the ILC

The e-e- running mode is one of the interesting physics options at the International Linear Collider (ILC). The luminosity for e-e- collisions is reduced by the beam-beam effects. The resulting beamstrahlung energy loss and beam-beam deflection angles as function of the vertical transverse offset are different compared to the e+e- collisions. In this paper, the dependence of these observables with the offset for different beam sizes has been analyzed to optimize performances for the e-e- mode, taking into account the requirements of the beam-beam deflection based intra-train feedback system. A first study of the implications for the final focus and extraction line optics is also presented for the cases of the 20 mrad and 2 mrad ILC base line crossing angle geometries

Plenty More Fish in the Sea? A working paper on the legal issues related to fishing beyond maximum sustainable yield: A UK case study

Maximum Sustainable Yield (MSY) is about to become a centre stage issue. The EU now has a commitment that fishing will progressively be managed at levels that correspond to MSY, and setting of quotas will respect scientific advice. Where possible, this is to be attained by 2015, and all fish stocks will be managed at MSY by 2020 at the latest. What the term ‘maximum sustainable yield’ means will dominate the next seven years of fisheries discussions. MSY has two different interpretations. Scientists use the term to mean the ability of a fish stock to support a sustainable fishing industry. Lawyers use it when assessing the obligations of a country in respect of its sovereign fishing rights. This report focuses on the legal interpretation of MSY. The report’s findings are particularly important as the EU has consistently over-allocated quota and ignored scientific advice, with the result that 75% of EU fish stocks are overfished. The starting point of a thorough investigation of MSY is the nature of fishing rights themselves. Fisheries were traditionally open access and seen as being inexhaustible. The origins of this approach date at least as far back as the era of the Roman Empire, but even then interpretation by the Roman courts created some perverse outcomes. The principle of open access became established for the English fishery in the 12th century. Before then, it is likely that all viable tidal fisheries were granted by the Crown into private hands following the Norman conquest. Indeed, some of these private tidal fisheries still persist today. In the seventeenth century, the mare liberum (freedom of the seas) debate confirmed open access on the high seas, but the position for territorial waters (within the 3 mile limit) was more complex. Under the common law, fishers operated under the public right to fish which permitted open access and has come to be interpreted as being based on the principle of inexhaustibility of the stock. Where this myth of inexhaustibility has come from is difficult to tell. It is often attributed to the Victorian scientist Thomas Huxley because of his pronouncement on the fecundity of some fish species, but Huxley himself highlights fisheries which were vulnerable to over-fishing and recognises that, on the high seas, no regulatory mechanisms were available, so this aspect of his findings was theoretical. Moreover, the negative impacts of some fisheries were known to the legislators at least since the Middle Ages and it is tempting to posit that public authorities, in whatever form they manifested themselves, have always been aware that some fishing methods can cause significant harm. As it stands, the basic principle the courts still adhere to is that the right to fish is untrammelled, but whether this principle would withstand direct judicial scrutiny is questionable. The United Nations Convention on the Laws of the Sea (UNCLOS) changed everything. Coastal states now enjoy sovereign rights over a far larger area of sea than they did previously. UK territorial waters now extend to 12 nautical miles and its exclusive economic zone (EEZ) extends up to 200 nautical miles. UNCLOS recognises sovereign rights for fisheries but also imposes duties on coastal states to restore stocks to levels which can produce MSY within their EEZs. Sovereign fishing rights are not untrammelled; they are qualified by a duty to restore stocks and to preserve the marine environment. Similar obligations are contained in the Convention on Biodiversity (with a target for 2020) and by virtue of the World Summit for Sustainable Development (with a target of 2015 for depleted stocks). Increasingly, international obligations of this nature are being enforced through the domestic courts. Many countries (including the UK itself for its Crown Dependencies and overseas territories) vest their fishing rights in an identifiable legal entity which then not just regulates the fishery, but also acts as an owner in the way it disposes of the fishery to commercial operators. Unfortunately, for the waters immediately adjacent to the UK, it is not clear in what Crown entity the UK’s fishing rights actually vest. The UK fishery is clearly some form of public asset and it is likely that there is a Crown trust in existence. The terms of this trust would place similar duties to maintain and restore fish stocks as those contained within UNCLOS. Despite a first instance decision to the contrary, it is possible that the terms of that trust are enforceable by third parties. There is a well-established principle in English law of nemo dat quod non habet (you cannot dispose of something you don’t own). The fact that the UK’s sovereign rights are qualified to MSY mean that the UK cannot grant to its fishers (or fishers of other Member States operating in UK waters) untrammelled rights which it does not possess itself. The same principle applies to the EU Common Fisheries Policy; the UK cannot confer on the EU greater fishing rights than those acknowledged by UNCLOS. The scientific origins of the term MSY were in the work of Beverton and Holt and related to a theoretical maximum, where a capture fishery will sustain itself. The legal meaning of the term gives discretion to fisheries managers to establish a management system which uses the best available science to determine what actions needs to be taken. The legal interpretation of MSY is therefore much broader than the scientific one and has the following features: • it is constantly evolving relating to the best available science of the time; • it defines the limits of a coastal state’s sovereign rights; • it may relate to the setting of fishing quota (where calculations of scientific MSY are particularly predominant), but many obligations for the attainment of MSY may also be satisfied by the imposition of technical or area-based management techniques such as marine protected areas; • it provides the framework in which fisheries should be managed; and • it can be used to show where stocks or areas of sea are not being properly managed, and thus direct where management resources should be targeted. The introduction of the ecosystem approach to fisheries management means that legal MSY can now take into consideration a much wider range of services provided by the fishery in the calculation and management of that fishery. UNCLOS itself permits taking into consideration relevant environmental and economic factors, including the economic needs of coastal fishing communities, but the ecosystem approach takes that a stage further by assessing all the services provided by the fishery. This shifts the focus away from simply treating the fishery as a commercial resource, to ensure that management benefits a much broader constituency. In 2008, the environmental non-governmental organisation (NGO), WWF tried to take the European Commission to the European Court of Justice for failing to allocate cod quota according to scientific advice. However, WWF was unable to access the European court system because the ECJ refused access to NGOs; ECJ rules at the time did not grant an NGO the right to be heard in court. Those rules are under scrutiny once again because they seem incompatible with the Aarhus Convention, so this position may change shortly. There are also other legal forums for testing the point. The UK courts do not have the same access restrictions as the ECJ, so a case launched in the UK courts concerning UK fishing rights or the voting patterns of the UK Minister in the EU could come to trial. The former President of the International Tribunal for the Laws of the Sea (ITLOS), Rüdiger Wolfrum, recommended ITLOS as having suitable jurisdiction to ensure compliance with MSY requirements The legal regime has dramatically moved away from the impossibility of mare liberum and open access fisheries to the possibility of effective regulation. Clearer sovereignty over the sea and better technology informing enforceable law means the legal context of fisheries have changed fundamentally for the better. There is an understanding that fisheries are held on trust for future generations. Fishing to MSY and trusteeship amount to the same thing; the legal interpretation of coastal states’ rights and responsibilities under MSY and the obligations of a Crown trust are the same. The ecosystem approach provides the matrix to understand who the beneficiaries of that trust are and where management effort needs to be directed. All these pieces are now in place and the hard law is there at the edges to ensure that reform remains on track. In short there are two key points to be drawn from this report: • It is illegal for coastal states to permit fishing beyond MSY; and • It is highly likely that this law is enforceable through the court system.Waterloo Foundatio

Husimi Transform of an Operator Product

It is shown that the series derived by Mizrahi, giving the Husimi transform (or covariant symbol) of an operator product, is absolutely convergent for a large class of operators. In particular, the generalized Liouville equation, describing the time evolution of the Husimi function, is absolutely convergent for a large class of Hamiltonians. By contrast, the series derived by Groenewold, giving the Weyl transform of an operator product, is often only asymptotic, or even undefined. The result is used to derive an alternative way of expressing expectation values in terms of the Husimi function. The advantage of this formula is that it applies in many of the cases where the anti-Husimi transform (or contravariant symbol) is so highly singular that it fails to exist as a tempered distribution.Comment: AMS-Latex, 13 page

Mutually unbiased bases: tomography of spin states and star-product scheme

Mutually unbiased bases (MUBs) are considered within the framework of a generic star-product scheme. We rederive that a full set of MUBs is adequate for a spin tomography, i.e. knowledge of all probabilities to find a system in each MUB-state is enough for a state reconstruction. Extending the ideas of the tomographic-probability representation and the star-product scheme to MUB-tomography, dequantizer and quantizer operators for MUB-symbols of spin states and operators are introduced, ordinary and dual star-product kernels are found. Since MUB-projectors are to obey specific rules of the star-product scheme, we reveal the Lie algebraic structure of MUB-projectors and derive new relations on triple- and four-products of MUB-projectors. Example of qubits is considered in detail. MUB-tomography by means of Stern-Gerlach apparatus is discussed.Comment: 11 pages, 1 table, partially presented at the 17th Central European Workshop on Quantum Optics (CEWQO'2010), June 6-11, 2010, St. Andrews, Scotland, U

Losses in the Post-Collision Extraction Line

The CLIC beam delivery system focuses 1.5 TeV electron and positron beams to a nanometre-sized cross section when colliding them at the interaction point (IP). The intense focusing leads to large beam-beam effects, causing the production of beamstrahlung photons, coherent and incoherent e+e− pairs, as well as a significant disruption of the main beam. The transport of the post-collision beams requires a minimal loss extraction line, with high acceptance for energy deviation and divergence. The current design includes vertical bends close to the IP in order to separate the charged particles with a sign opposite to the main beam into a diagnostic-equipped intermediate dump, whilst transporting the photons and the main beam to the final dump. Photon and charged particle losses on magnet masks and dumps result in a complex radiation field and IP background particle fluxes. In this paper, the electromagnetic backgrounds at the IP arising from the losses occurring closest to the collision point are calculated

Photon backgrounds at the CLIC interaction point due to losses in the post-collision extraction line

The CLIC beam delivery system focuses 1.5~TeV electron and positron beams to a nanometre-sized cross section when colliding them at the interaction point (IP). The intense focusing leads to large beam-beam effects, causing the production of beamstrahlung photons, coherent and incoherent $e^+e^-$ pairs, as well as a significant disruption of the main beam. The transport of the post-collision beams requires a minimal loss extraction line, with high acceptance for energy deviation and divergence. The current design includes vertical bends close to the IP in order to separate the charged particles with a sign opposite to the main beam into a diagnostic-equipped intermediate dump, whilst transporting the photons and the main beam to the final dump. Photon and charged particle losses on magnet masks and dumps result in a complex radiation field and IP background particle fluxes. In this paper, the electromagnetic backgrounds at the IP arising from the losses occurring closest to the collision point are calculated