William H. Klink Department of Physics and Astronomy

 Research Interests
 Most of my research interests deal with the application of symmetry to quantum theory and quantum mechanical systems. Some of my work deals with the analysis of groups, which provide the mathematical expression of symmetry. Applications include using symmetry to develop a relativistic quantum theory for systems with finite and infinite degrees of freedom.
 More recently I have been working on using symmetry to show why quantum theory has the structure that is does. This has led to work dealing with the interpretation of quantum theory, as well as work in the field of science and religion.
 In what follows I divide previous and ongoing work into six different categories.
 I. Relativistic Quantum Theory
 There are a number of ways of incorporating the symmetry expressed in Einstein relativity with the quantum theory for finite degree of freedom systems. I have worked on a version called point form relativistic quantum theory, in which relativistic systems interact via a mass operator to produce bound states and scattering. A notable property of the point form is that it is manifestly Lorentz covariant, meaning that it is easy to Lorentz transform quantum systems. Details are given in reference (1).
 While on leave at the Institute for Theoretical Physics in Graz, Austria (academic year 19992000), I collaborated with Professor Willi Plessas and his coworkers on applying point form quantum theory to the electroweak properties of nucleons, viewed as bound states of three quarks. We were able to get good results for the socalled form factors of both the proton and neutron (2).
 Using similar techniques, my graduate student Tom Allen and I, along with Professor Wayne Polyzou, calculated the form factors of the deuteron (3); the results, however, were not as good as those for the nucleons. This has led to further collaboration with Professor Wolfgang Schweiger, also at the Institute for Theoretical Physics in Graz, in developing more refined models for calculating formfactors of hadrons, as well as investigating their decay properties (4).
 II. Point Form Quantum Field Theory
 Quantum Field Theory combines the general structure of quantum theory with the symmetry expressed in Einstein relativity for infinite degree of freedom systems. In the point form version of quantum field theory, all interactions between systems are contained in the energy and momentum operators. A first paper on point form quantum field theory for spin zero and ˝ particles was done with Professor Schweiger and his students (5). Reference (6) summarizes results for both versions of point form quantum theory.
 At present we are working on adapting the point form to wellknown solvable models called the Thirring and Schwinger models, as well as generalizing these models to four dimensional spacetime.
 III. Representation Theory of the Unitary Group
 There are a number of different types of groups that express the notion of symmetry, depending on the quantum system under consideration. One of the fundamental properties of a quantum theory is that states of a quantum system are described by elements of a vector space, so that states can be superposed. Moreover groups can be represented as operators acting on a vector space called the representation space. That is, a given group generates a class of representation spaces that are the vector spaces of interest for given quantum systems.
 One of the most important of the groups whose representation spaces can be applied to quantum systems are the unitary groups. The many applications of these groups and their representations are sketched in the introduction to reference (7). For many years I have collaborated with Professor Tuong TonThat, in the mathematics department at the University of Iowa, on the decomposition of tensor products of representations of the unitary groups (7a). We have developed algorithms for computing socalled ClebschGordan and Racah coefficients for the unitary groups. This work culminated in three publications, one a review of the underlying theoretical structure of these coefficients (7); the other two papers deal with implementing the algorithms on computers (8). The computer programs for implementing the algorithms were largely developed by Steven Gliske, an undergraduate advisee at the time, and can be found on my Website.
 Another important class of groups are those that leave a spacetime manifold invariant. The best known are the Galilei group and the Poincare group, to be discussed in more detail in section V. The irreducible representations of these groups are wellknown, and are interpreted as giving oneparticle properties of quantum systems. Here also tensor products of representations are important, for they provide the description of manyparticle systems. I have worked on the decompositions of such tensor products into direct sums (and integrals) of irreducible representations, and computed the ClebschGordan and Racah coefficients that pertain (9).
 IV. Acceleration Groups, Fictitious Forces, and Gravity
 In classical mechanics it is wellknown that accelerating reference frames lead to fictitious forces, forces that arise only because the physics is being done in a noninertial reference frame. A simple way to see how fictitious forces arise is by using Lagrange's equations, which are covariant with respect to arbitrary coordinate transformations. By transforming from an inertial frame to an accelerating frame, Lagrange's equations automatically lead to fictitious forces. Such forces also lead to gravitational forces through the Equivalence Principle, a cornerstone of Einstein's theory of gravitation.
 Fictitious forces also arise in quantum theory. For nonrelativistic quantum theory, the group that connects different inertial frames is called the Galilei group. As discussed in the next section its representations largely determine the structure of nonrelativistic quantum theory. Transformations to accelerating reference frames lead to a larger group called the Euclidean line group, whose elements are maps from the real line (for the time parameter) to the (Euclidean) group consisting of rotations and velocity transformations.. By requiring that the representations of the Euclidean line group act on the same space as the physical representations of the Galilei group, it is possible to show how the time dependent Schroedinger equation leads to linear fictitious forces(10).
 What is still missing are rotational fictitious forces, such as centripetal forces. I am collaborating with Professor Sujeev Wickramasekara from Grinnell College (Grinnell, Iowa) on generalizing socalled cocycle representations for time dependent rotations. Moreover, for more general coordinate transformations there is a connection between fictitious forces and gravitational forces, a manifestation of the Equivalence Principle for nonrelativistic systems.
 We also show that it is possible to violate the Equivalence Principle in nonrelativistic quantum theory (11).
 V. Foundations of Quantum Theory
 A requirement on any physical theory is that it should be formulated in such a way as to be valid in any inertial reference frame. Classical Newtonian mechanics satisfies this requirement. Such theories are called covariant with respect to a group of transformations, and the group itself is called the relativity group. There are three relativity groups that at present are of importance in physics. The first and most intuitive is Newton relativity, in which the relativity group is called the Galilei group. The elements of the Galilei group are rotations, spatial and temporal translations and Galilei boosts, transformations between different inertial frames moving at constant velocity with respect to one another.
 However, Newton relativity is inadequate for velocities close to the speed of light. Then Newton relativity is replaced by Einstein relativity and Galilei boosts are replaced by Lorentz boosts. With this change the relativity group is called the Poincare group. Finally when the curvature of spacetime becomes relevant, a new relativity called deSitter relativity with an accompanying group called the deSitter group is required.
 Each of the relativity groups generate a spacetime manifold. Thus the Galilei group generates our intuitive notion of space called Euclidean space, while the Poincare group generates the spacetime manifold called Minkowski space(time). For deSitter relativity the manifold is called deSitter space.
 There is a quantum theory associated with each relativity type. The best established is based on Newton relativity and is called nonrelativistic quantum theory. The relativistic theories discussed in sections I and II are based on Einstein relativity; deSitter quantum theory is only now being developed.
 The basic structure of each of the quantum theories is generated from the irreducible representations of the corresponding relativity group. Moreover, the vector spaces needed for a quantum theory are just the irreducible representation spaces of the respective groups. And each type of irreducible representation provides a classification for possible types of matter. For Newton relativity the only possible type of matter is that characterized by mass and spin, whereas for Einstein relativity, new types of matter besides those described by mass and spin (for example, the electron, with mass .5 Mev and spin 1/2) are possible.
 The best known new type of matter is a class of massless particles such as photons or gluons; but if indeed particles are found that are traveling at speeds greater than the speed of light, they too would constitute a new type of matter (called tachyons) whose structure would be given by the irreducible representations of the Poincare group. Finally, deSitter representations provide another classification of types of matter, possibly associated with dark matter; this is at present being investigated.
 Many of the wellknown structural features of quantum theory are a consequence of this symmetry approach to quantum theory. For example, in nonrelativistic quantum theory, the Heisenberg uncertainty relations, as well as the necessary quantization of angular momentum are consequences of the representation properties of the Galilei group. Professor Wickramasekara and I are presently writing a detailed account of how symmetry provides a grounding principle for quantum theory.
 Another issue in the foundations of quantum theory has to do with the interpretation of the theory. One of the most amazing and confounding features of quantum theory is how measurements bring properties of a system into being. Before a measurement a quantum system may be described by a state vector that does not allow for a specific property such as spin projection; but a (for example, SternGerlach) measurement then brings that property into being. What seems to be happening is that the measurement apparatus provides the quantum system with a set of alternatives, one of which is realized after the measurement has taken place. How does this happen? While a variety of explanations have been offered, I have developed a model in which it is the quantum system itself that opts, out of the set of alternatives, that which is actually realized. Reference (12) provides more details and context for such a model, which says in effect that quantum systems exhibit elementary manifestations of freedom.
 VI. Science and Religion
 I have, for many years, been teaching and collaborating with Professor David Klemm, of the Department of Religious Studies at the University of Iowa. One of the goals of our collaboration has been to show that theology need not be only a confessional enterprise, but rather a discipline with its own distinctively cognitive structure. We have analyzed the role of models in scientific thinking and shown how the notion of modeling provides a way of constructing and testing theological models (13). This work has led to a study of the notion of freedom as it appears in a hierarchy of matter, stretching from quantum systems through organic matter up to human beings. (See reference (14))
 We are at present working through the consequences of a thoroughgoing dualism, with the material world on the one hand, and what we call the transcendent order on the other. Physical theories such as nonrelativistic quantum theory are not themselves material. They reside, along with mathematics, logic, and more generally abstract thinking, in a transcendent order that is as real as the material world. Some of the preliminary consequences of such a dualism can be found in reference (15).
 References
 1. W. KLINK, Point Form Relativistic Quantum Mechanics and Electromagnetic Form Factors, Phys. Rev. C 58 (1998) 3587; W. KLINK, Relativistic Simultaneously Coupled Multiparticle States Phys. Rev. C 58 (1998) 3617; W. KLINK, Constructing Point Form Mass Operators from Vertex Interactions, Nucl Phys A716 (2003) 123.
 2. R. WAGENBRUNN, S. BOFFI, W. KLINK, W. PLESSAS and M. RADICI, Convariant Nucleon Electromagnetic Form Factors from the Goldstonebosonexchange Quark Model, Phys. Lett. B 511 (2001) 33; L. YA. GLOZMAN, M. RADICI, R. WAGENBRUNN, S. BOFFI, W. KLINK, and W. PLESSAS , Covariant Axial Form Factor of the Nucleon in a Chiral Constituent Quark Model , Phys. Lett. B 516 (2001) 183; S. BOFFI, L. Ya. GLOZMAN, W. KLINK, W. PLESSAS, M. RADICI, and R. WAGENBRUNN , Covariant Electroweak Nucleon Form Factors in a Chiral Constituent Quark Model, Eur. Phys. J. A14 (2002) 17.
 3. T. ALLEN, W. KLINK and W. POLYZOU, PointForm Analysis of Elastic Deuteron Form Factors, Phys. Rev. C 63 (2001) 034002; W. KLINK, Structure and Applications of Point Form Relativistic Quantum Mechanics, in Proceedings of the XVIII European Conference on FewBody Problems in Physics, Bled, Slovenia, Few Body Systems, Supplement 14 (2003) 387.
 4. E. BIERNAT, W. KLINK, W. SCHWEIGER, A Relativistic Coupled Channel Formalism for Electromagnetic Form Factors of Two Body Bound States, FewBody Systems 50 (2011) 435; E. BIERNAT, W. SCHWEIGER, K. FUCHSBERGER, W. KLINK, Electromagnetic Meson Form Factor from a Coupled Channel Approach, Phys. Rev. C79 (2009) 055203.
 5. E. BIERNAT, K. FUCHSBERGER, W. KLINK, W. SCHWEIGER, Point Form Quantum Field Theory, Ann. Phys. 323 (2008) 1384.
 6. E. BIERNAT, W. KLINK, W. SCHWEIGER, Point Form Hamiltonian Dynamics and Applications, FewBody Systems, 49 (2011) 149.
 7. W. KLINK, T. TONTHAT, Invariant Theory of Tensor Product Decompositions of U(N) and Generalized Casimir Operators, Notices of the AMS, 56 (2009) 931;
 7a. W. KLINK, T. TONTHAT, Invariant Theory of the Block Diagonal Subgroups of GL(N,C) and Generalized Casimir Operators, J. Alg. 145 (1992)187; W. KLINK and T. TONTHAT, Representations of Sn × U(N) in Repeated Tensor Products of the Unitary Group, J. Phys. A: Math. Gen. 23 (1990) 2751; W. KLINK, SU(3) ClebschGordan Coefficients with Definite Permutation Symmetry Ann. Phys. 213 (1992) 54; W. KLINK and T. TON THAT, Multiplicity, Invariants and Tensor Product Decompositions of Compact Groups, J. Math. Phys. 37 (1996) 6468.
 8. S. GLISKE, W. KLINK, and T. T0NTHAT, Algorithms for Computing U(N) ClebschGordan Coefficients, Acta Applicande Mathematica, 95 (2007) 51; S. GLISKE, W. KLINK and T. TONTHAT Algorithms for Computing Generalized U(N) Racah Coefficients, Acta. Applicande Mathematica 88, (2005) 229249.
 9. W. KLINK, ClebschGordan and Racah Coefficients of the Poincaré Group Ann. Phys. 213 (1992) 31.
 10. W. KLINK, Quantum Mechanics in Noninertial Reference Frames I: Nonrelativistic Quantum Mechanics, Ann. Phys. (NY) 260 (1997) 27; W. KLINK, Quantum Mechanics in Noninertial Reference Frames and Representations of the Euclidean Line Group, in Proceedings of the Second International Conference on Symmetry in Nonlinear Mathematical Physics, edited by M. Shkil, A. Nikitin, and V. Boyko, Institute of Mathematics of the National Academy of Sciences of Ukraine, 1997, p. 254.
 11. W. KLINK, S WICKRAMASEKARA, Quantum Mechanics in Noninertial Reference Frames II: Violations of the Nonrelativistic Equivalence Principle (to be submitted)
 12. W. KLINK, Dualism and Quantum Theory: Freedom and Matter (to be submitted)
 13. D. KLEMM, W. KLINK, Constructing and Testing Theological Models, Zygon: The Journal of Religion and Science, 38 (2003) 495.
 14. D. KLEMM, W. KLINK, Consciousness and Quantum Mechanics; Opting from Alternatives, Zygon, Journal of Religion and Science, 43 (2008) 307.
 15. D. KLEMM, W. KLINK, Freedom and Matter (to be published)
 Related Web Sites
 UI Theoretical Nuclear Physics
Programs for Computing Generalized U(N) Racah Coefficients