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ScienceWeek
THEORETICAL PHYSICS: ON LIMITATIONS
The following points are made by Frank Wilczek (Physics Today 2004 January):
1) There have been extraordinary triumphs of physicists using analysis and synthesis or, alternatively, reductionism, to account for the behavior of matter and the structure of the universe as a whole. However, there has been quite a lot that physicists might once have hoped to derive or explain based on fundamental principles, for which that hope now seems dubious or forlorn. One important limitation concerns the lack of a principle that could lead to a unique choice among different seemingly possible solutions of the fundamental equations and could select out the Universe we actually observe.
2) It is very instructive to consider the corresponding problem for atoms and matter. Like the classical equations governing planetary systems, the equations of quantum mechanics for electrons in a complex atom allow all kinds of solutions. In fact, the quantum equations allow even more freedom of choice in the initial conditions than the classical equations do. The wavefunctions for N particles live in a much larger space than the particles do: They inhabit a full-bodied 3N-dimensional configuration space, as opposed to 2N copies of three-dimensional space. (For example, the quantum description of the state of two particles requires a wavefunction that depends on six variables, whereas the classical description requires 12 numbers, namely their positions and velocities.)
3) Yet the atoms we observe are always described by the same solutions -- otherwise we would not be able to do stellar spectroscopy, or even chemistry. Why? A proper answer involves combining insights from quantum field theory, mathematics, and some cosmology. Quantum field theory tells us that electrons --unlike planets -- are all rigorously the same. Then the mathematics of the Schroedinger equation, or its refinements, tells us that the low-energy spectrum is discrete, which is to say that if our atom has only a small amount of energy, then only a few solutions are available for its wavefunction.
4) But because energy is conserved, this explanation begs another question: What made the energy small in the first place? Well, the atoms we study are not closed systems; they can emit and absorb radiation. So the question becomes: Why do they emit more often than they absorb, and thereby settle down into a low-energy state? That's where cosmology comes in. The expanding Universe is quite an effective heat sink. In excited atoms, energy radiated as photons eventually leaks into the vast interstellar spaces and redshifts away. By way of contrast, a planetary system has no comparably efficient way to lose energy -- gravitational radiation is ridiculously feeble -- and it cannot relax.
5) So one selection principle that applies to many important cases is to choose solutions with low energy. In the same spirit, when the residual energy cannot be neglected, one should choose thermal equilibrium solutions. This is appropriate for systems and degrees of freedom that have relaxed, but is not appropriate in general. The selection procedure that dominates the literature of high-energy physics and string theory is energy based. It is traditional to identify the lowest-energy solution with the physical vacuum, and to model the content of the world using low-energy excitations above that state. For the solution to count as successful, the excitations should include at least the particles of the standard model.
Physics Today http://www.physicstoday.org
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ON FIELD THEORY IN PHYSICS
In physics, a field is an entity that acts as intermediary in interactions between particles, and which is distributed over part or all of space, and whose properties are functions of space coordinates, and except for static fields, also functions of time. There is also a quantum-mechanical analog of this entity, in which the function of space and time is replaced by an operator at each point in space-time.
The following points are made by Roman Jackiw (Proc. Natl. Acad. Sci. US 1998 95:12776):
1) Present-day theory for fundamental processes (i.e., descriptions of elementary particles and forces) is phenomenally successful. Experimental data confirms theoretical prediction, and where accurate calculation and experiments are attainable, agreement is achieved to 6 or 7 figures. Two examples: a) The helium atom ground state energy (*Rydbergs) is experimentally measured as -5.8071394 and theoretically calculated as -5.8071380. b) The muon magnetic dipole moment is experimentally measured as 2.00233184600 and theoretically calculated as 2.00233183478.
2) The theoretical structure within which this success has been achieved is *local field theory, which offers a wide variety of applications, and which provides a model for fundamental physical reality as described by our theories of *strong, electroweak, and gravitational processes. No other framework exists in which one can calculate so many phenomena with such ease and accuracy.
3) But is spite of these successes, today there is little confidence that field theory will advance our understanding of nature at its fundamental workings beyond what has already been achieved. Although in principle all observed phenomena can be explained by present-day field theory, these accounts are still imperfect, requiring ad hoc inputs. Moreover, because of conceptual and technical obstacles, classical gravity theory has not been integrated into the *quantum field description of nongravitational forces: *quantizing the *metric tensor of Einstein's theory produces a quantum field theory beset by infinities that apparently cannot be controlled.
4) These shortcomings are actually symptoms of a deeper lack of understanding concerning *symmetry and symmetry breaking... Physicists are happy in the belief that Nature in its fundamental workings is essentially simple, but observed physical phenomena rarely exhibit overwhelming regularity. Therefore, at the very same time that we construct a physical theory with intrinsic symmetry, we must find a way to break the symmetry in physical consequences of the model.
5) These problems have produced a theoretical impasse for over two decades, and in the absence of new experiments to channel theoretical speculation, some physicists have concluded that it will not be possible to make progress on these questions within field theory, and they have turned to a new structure, "*string theory". In field theory, the quantized excitations are point particles with point interactions, and this gives rise to the infinities. In string theory, the excitations are extended objects -- strings -- with nonlocal interactions; there are no infinities in string theory, and that enormous defect of field theory is absent.
6) Yet in spite of its positive features, until now string theory has provided a framework rather than a definite structure, and a precise derivation of the *Standard Model has yet to be given. The author concludes: "On previous occasions when it appeared that quantum field theory was incapable of advancing our understanding of fundamental physics, new ideas and new approaches to the subject dispelled the pessimism. Today we do not know whether the impasse within field theory is due to a failure of imagination or whether indeed we have to present fundamental physical laws in a new framework, thereby replacing the field theoretic one, which has served us well for over 100 years."
Proc. Nat. Acad. Sci. http://www.pnas.org
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Notes:
Rydbergs: A unit of energy used in atomic physics, value = 13.605698 electronvolts.
local field theory: In this context, "locality" is the condition that two events at spatially separated locations are entirely independent of each other, provided that the time interval between the events is less than that required for a light signal to travel from one location to the other. For example, the quantum mechanical wave function is a "local" field.
strong, electroweak, and gravitational processes: The fundamental forces comprise the gravitational force, the electromagnetic force, the nuclear strong force, and the nuclear weak force. The "electroweak" interactions are a unification of the electromagnetic and nuclear weak interactions, and are described by the Weinberg-Salam theory (sometimes called "quantum flavordynamics"; also called the Glashow-Weinberg-Salam theory).
quantum field description: In general, a quantum field theory is a quantum mechanical theory applied to systems having an infinite number of *degrees of freedom.
degrees of freedom: In general, the number of independent parameters required to specify the configuration of a system.
quantizing: In experimental physics, a quantized variable is a variable taking only discrete multiple values of a quantum mechanical constant. In theoretical physics, "quantizing" means the consistent application of certain rules that lead from classical to quantum mechanics. In general, "quantization" is a transition from a classical theory or a classical quantity to a quantum theory or the corresponding quantity in quantum mechanics.
metric tensor: The mathematical statement (involving a set of quantities) that describes the deviation of the Pythagoras theorem in a curved space.
symmetry and symmetry breaking: If a theory or process does not change when certain operations are performed on it, the theory or process is said to possess a symmetry with respect to those operations. For example, a circle remains unchanged under rotation or reflection, and a circle therefore has rotational and reflection symmetry. The term "symmetry breaking" refers to the deviation from exact symmetry exhibited by many physical systems, and in general, symmetry breaking encompasses both "explicit" symmetry breaking and "spontaneous" symmetry breaking. Explicit symmetry breaking is a phenomenon in which a system is not quite, but almost, the same for two configurations related by exact symmetry. Spontaneous symmetry breaking refers to a situation in which the solution of a set of physical equations fails to exhibit a symmetry possessed by the equations themselves.
string theory: In particle physics, string theory is a theory of elementary particles based on the idea that the fundamental entities are not point-like particles but finite lines (strings), or closed loops formed by strings, the strings one-dimensional curves with zero thickness and lengths (or loop diameters) of the order of the Planck length of 10^(-35) meters.
Standard Model: In particle physics, the "Standard Model" is a theoretical framework whose basic idea is that all the visible matter in the universe can be described in terms of the elementary particles leptons and quarks and the forces acting between them. Leptons are a class of point-like fundamental particles showing no internal structure and no involvement with the strong forces. Electrons and neutrinos are among the particles classified as leptons. The strong force (nuclear strong force) is one of the four fundamental forces: the gravitational force, the electromagnetic force, the nuclear strong force, and the nuclear weak force (see below), with the strong force approximately 100 times stronger than the electromagnetic force. A quark is a hypothetical fundamental particle, having charges whose magnitudes are one-third or two-thirds of the electron charge, and from which the elementary particles may in theory be constructed. At the present time, ongoing experimental projects in particle physics are expected to permit a completion of the Standard Model, but a unified theory of all forces known to physics is not yet in sight.
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