Mathematical Physics

Mathematical physics Scientific  discipline which aims to describe physical phenomena in rigorous mathematical terms.

IN-DEPTH ABSTRACTfrom Mathematical Physics by Gianfausto Dell’Antonio (Encyclopedia of Science and Technology)

Research in mathematical physics is divided into three phases, which characterize this discipline and distinguish it from other branches of physics and mathematics. In the first phase we proceed to the construction of a model that manages to represent the aspects that are considered important in the physical phenomenon under consideration. A model is therefore a structure, described in mathematical terms, whose purpose is to make bodies of mathematical language correspond to the components of a physical phenomenon and their relationships. For example, a model of the Earth-Moon system can consist of two geometric points in three-dimensional space, to which two parameters (the masses of the related celestial bodies) and the force with which they attract are associated. The model is not a faithful representation of reality, for example in the present case the extent and composition of the celestial bodies are not taken into consideration, but provides a first approximation. More refined models can be considered later, for example by treating Earth and Moon as geometric solids with variable density.

In the construction of a model we generally start from simple structures, to modify them subsequently, seeking a compromise between the difficulty of the mathematical problems posed by the model and the accuracy with which this represents the physical system. By definition, no model describes all aspects of reality, and the mathematical physicist must be able to grasp the significant components in the phenomenon he wants to analyze. The second phase of the research consists in ‘solving’ the model that has been built. Since this (rather than reality) is treated in mathematical terms, using structures that can also be very different from each other, the solution of the model can from time to time mean quite different things. Usually mechanical models, relating for example toSolar system, the mechanics of fluids, the deformation of elastic bodies or the behavior of bodies with electric charge in an electromagnetic field, comprise two main components: the possible states of the system and the evolution equation. In this case, solving the model means demonstrating that, at each initial datum, there is one and only solution of the evolution equation and possibly describing, at least qualitatively, its properties. In other models, such as those adopted in statistical mechanics or in the description of the balance of elastic bodies, in order to obtain the solution an attempt is made to identify a configuration, not necessarily unique, which minimizes a quantity that is considered empirically relevant, for example the energy, entropy or surface energy. In this second phase, the mathematical physicist uses tools of various nature, belonging to different fields of mathematics: analysis, algebra, geometry, numerical analysis, calculation of probabilities, etc .; on many occasions it is located inhaving to “/> having to modify or improve these tools and sometimes to build new ones, more suitable for the study of the specific problem. The final phase of research in mathematical physics consists in using the mathematical results obtained to deduce estimates and predictions on the behavior of the system physical under examination.

This subdivision into three phases is also common to other disciplines that aim to describe physical phenomena, but in research fields other than mathematical physics the logical-deductive mathematical structure of the second phase often gives way to at least partly heuristic and conjectural considerations. It is also true that even in the activity of a mathematical physicist these certain predictions, in the form of theorems, are generally preceded by a phase in which intuition, the analysis of elementary models, analogies and preliminary calculations play a determining role (in this, electronic calculators have played a major role in recent times). Only in a second moment, in light of the qualitative considerations made, theorems are enunciated and proved.

  1. From quantum mechanics to quantum field theory

Non-relativistic quantum mechanics, although adequate to describe the atomic and molecular structure, appears insufficient to describe the interactions between particles and electromagnetic field. The photoelectric effect, combined with the wave-particle dualism, leads to associate the electromagnetic field with photons, particles of zero mass, the number of which is susceptible to variation due to the interaction with matter. This makes Schrödinger’s formalism, in which the number of particles is constant, unsuitable for the treatment of the electromagnetic field and identifies the functional space in which to write the equations, i.e. the space of square functions integrable on R n , if consider a system of nspatial dimension particles 3. In vacuum, each component of the electromagnetic field satisfies the wave equation; if the spatial dimension is 1, periodic conditions relating to the segment of extremes 0-L are imposed and the Fourier series development is used, the wave equation appears as an equation for infinite harmonic oscillators, each with frequency nL -1 , having chosen a unit of measurement system / “> measure in which the speed of light is equal to 1. The evolution equation for each of these oscillators is Hamiltonian and the corresponding quantum Hamiltonian, obtained with the quantization rules /”> quantization Heisenberg and Schrödinger, has the polynomials of Hermite as eigenfunctionsn and multiple eigenvalues ​​of nL  1 (unless a translation by a constant).

It is then possible to interpret the state n as a state with n particles, the photons of energy nL  1 , and the consequent description of the Hilbert space associated with the ‘quantized’ electromagnetic field as a direct sum of spaces, parameterized by an index n integer, each described in terms of the number of occupation of photons of frequency nL  1 and assigned energy. This description, called the representation of Fock, by Vladimir A. Fock which suggested it in this form in the mid-thirties of the twentieth century, is called second quantization, where first is the formulation of the quantum mechanics of Heisenberg and Schrödinger. The word quantization is associated with this formulation since the quantum aspects it describes are connected (although not from an analytical point of view) to the fact that the energy levels of the atoms are quantized.

Fock’s representation, placed in a mathematically more precise form by Valentine Bargmann and especially by Irving E. Segal in the early 1960s, is at the basis of the theory of quantized fields. In the exposition we have given, it was assumed that the space was one-dimensional and periodic conditions were set at the ends of a segment. The formulation can be easily generalized to the case of R 3. In this representation, the creation and destruction operators associated with a degree of freedom play an important role, the action of which is to move the system from one state to another in which the number of photons has increased or decreased by one unit. The operators representing the field and its derivative with respect to time are obtained by the sum and difference of the creation and destruction operators, with coefficients able to reproduce the field’s covariance properties and to ensure that the field itself satisfies Maxwell’s equations ( which, remember, are hyperbolic equations and therefore require for the resolution the data of the field and its time derivative in a predetermined instant).

This interpretation allows to study interactions in which the number of photons is not conserved and therefore to analyze phenomena such as the photoelectric effect. The interaction of matter, in its non-relativistic aspects, with the quantized electromagnetic field, is now described in terms of a Hamiltonian sum of a free part (the sum of the non-interacting Hamiltonians of the field and particles respectively) and of a Hamiltonian d interaction, which in general is chosen linear in the field creation or destruction operators, with a coefficient proportional to the operator representing the current (a construction that corresponds to the quantization of the interaction of the classic electromagnetic field with the currents). Since the proportionality coefficient is the electric charge e, which in natural units is very small (1/137 ca.), one can think of adopting a perturbative method for solving problems, although, as a consequence of the singular character of the interaction at high frequencies (at small distances), the perturbation series generally present formal difficulties, in the form of divergences of integrals, which must be resolved with appropriate artifices (which go by the name of renormalization) for each order in and . Considerable progress has recently been made in a non-perturbing treatment of the theory, in particular regarding simplified models (Pauli-Fierz models), but a complete treatment has not yet been elaborated.

The perturbative treatment of relativistic quantum electrodynamics has led to predictions that have been confirmed by experimental data with an exceptional degree of precision and some theoretical difficulties, connected to the ambiguity in the product of distributions, have been overcome thanks to an accurate microlocal analysis of wave fronts. It seems reasonable to think that, in the long term, it is possible to give a complete mathematical treatment, even if this has not been done so far. On the other hand, the perturbation series does not meet any of the convergence criteria, even weak (Borel convergence), so it is currently unclear whether this series corresponds to a well-defined solution. The perturbation theory, within a quantum field theory, it has had a remarkable development since the 1950s, with important contributions, among others, by Julian Schwinger, Sin-Itiro Tomonaga, Freeman Dyson, Nikolai N. Bogoljubov and especially Richard P. Feynman; Feynman’s prescription in terms of diagrams for calculating the values ​​of the fields and the matrixS (which describes the scattering phenomena ) expected in vacuum is still the basis of most of the works in this sector and has produced results of considerable value, also from a mathematical point of view.

Following the success, at least perturbative, of quantum electrodynamics and on the basis of the wave-particle duality principle, for the study of elementary particles, theories that introduce fields associated with each particle and an interaction deduced by quantizing the particle fields, along the lines of the electromagnetic field quantization. They are therefore introduced also for the operating particles of creation and destruction, and a suitable state of vacuum, corresponding to the zero occupation number for all the particles. The fields introduced are of various nature: scalar, vector or in any case with full or spinor spin (for semi-integer spin particles, the latter correspond to irreducible representations of equal size of the SU group (2)). Their quantization uses, similarly to what has been done for the electromagnetic field, canonical switching relations for whole spin particles, corresponding to switching relations for fields in spatially separated regions at a fixed time, or to canonical anticommutation relations for semi-integer spin particles, corresponding to fields that anticommute. For the latter, it is easy to show that the occupation number for each degree of freedom can only take the values ​​0 or 1, in accordance with the Pauli exclusion principle.

In the description provided by the theory of quantized fields of strong interactionsbetween elementary particles, the coupling constants are very large, of the order of magnitude of 10 in natural units (although the definition of the coupling constant itself is uncertain, due to the ambiguities that a rigorous treatment presents) and the predictive value of a disturbing series is uncertain for such large values ​​of the development parameter. On the other hand, until the early seventies of the twentieth century the only example of rigorously formulated quantum field theory was free field theory, interesting from a mathematical point of view but of little physical relevance. This theory was built by Kurt Otto Friedrichs and Segal in the 1950s, and was defined on the Fock space. We give here some details for the case of the scalar field of mass m. The field f ( x ) is constructed as an operator-valued distribution, which satisfies the equation (; 2 1 2 ) f ( x ) 50, where; 2  is the Laplace operator. The f ( x ) field also satisfies the (canonical) switching relationships [f ( x ), f ( x 9)] 5 iD ( x 2 x 9), x 5 { 0 5 ct , 1 , 2 , 3 }, where D ( y) is the real solution of the equation; D 50 which is canceled outside the light cone. From the equation that satisfies it is deduced that the field itself can be written as the sum of a part with a positive frequency and a part with a negative frequency. This field can be realized in a Fock space that has a state w (the void) ‘annihilated’ by the negative frequency part, while the polynomials in the positive frequency part generate adense togetherif applied to w. The equations satisfied by the field are Hamiltonian and the corresponding energy is positive, and is canceled only at the void.

This construction can be considered the Fock quantization of the symplectic dynamics defined on the linear variety of the classic solutions of the wave equation. In order to deal with a theory with interaction, Segal attempted, but without success, to quantify the symplectic dynamics defined on the (no longer linear) variety of solutions of the wave equation with a termnot linear. In the same period Arthur S. Wightman formulated an axiomatic theory for relativistic quantum fields, based on the hypotheses of location, existence of the void and positivity of energy (defined as the generator of the unitary group that describes the translations over time). As a consequence of the axioms, the functions W ( 1 ,  , n ) 5 (w, f ( 1 ), f ( n) w) (Wightman functions) have properties of invariance, positivity (consequence of the positivity of the Hilbertian scalar product) and analyticity (consequence of the positivity hypothesis of the energy spectrum). Wightman also showed that the field can be completely ‘rebuilt’ when the W functions are known ( 1 ,  , n ).

A similar axiomatic theory was shortly after suggested by Rudolph Haag, making use of algebras of limited operators associated with each limited region of space-time, with natural conditions of inclusion and of commutativity at ‘spatial’ distances. Haag also postulated the existence of a generator of the group of temporal translations, with spectrum contained in the positive half-axis and an eigenvalueand it is possible to formulate an axiomatic theory also for Schwinger’s functions. Konrad Osterwalder and Robert Schrader have shown that every Euclidean theory thus formulated corresponds in a univocal way to a Minkowskian theory of Wightman and therefore the construction of a relativistic field theory can be developed by explicit construction of a Schwinger function.

These functions have specific positivity characteristics, highlighted above all by Kurt Symanzik, which make it possible to interpret them as transition functions for a stochastic process. Following these interpretations, in 1971 Edward Nelson managed to build a stochastic process in R 1 with which Schwinger functions are associated which, prolonged in the complex plane, give rise to Wightman functions satisfying the axioms. The corresponding relativistic quantum field is defined on a minkowskian space-time of dimension two and satisfies the equation (; 2 1 2 ) F ( x ) 5: P (F ( x )): where Pis a polynomial of odd degree with coefficient of the term of higher negative degree and the symbol:: indicates that the distributions do not form an algebra, so their product must be interpreted appropriately. In the previous equation, the field: P (F ( x )): is local.

Following a different procedure, a year later JamesGlimm and Arthur Jaffe built the same field using statistical mechanics methods, but working directly in Minkowski’s space. Considerably generalizing the method followed by Glimm and Jaffe, a few years later Jacques Magnen and Roland Seneor managed to build a quantum theory in a Minkowski space of dimension 3 (and therefore of spatial dimension 2), which is the quantization of a classical theory associated with the equation (; 2 1 2 ) f ( x ) 5 g : f 3 ( x ): with g constant negative (called theory f 4for the formal expression of the corresponding potential). For polynomials of degree greater than three, and for a Minkowski space of dimension four, the theory has not yet been constructed and there are indications that the quantum theory f 4 is not constructible, at least with the methods used so far, due to the excessive singularity of terms. This would make the rigorous construction of a quantum gauge theory more interesting.

 

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