In contemporary field theory, the word classical is reserved for an analytical framework in which the local equations of motion provide a complete description of the evolution of the fields. Classical field theory is a differential expression of change in functions of space and time, which summarizes the state of a physical system entirely in terms of smooth fields. The differential (holonomic) structure of field theory, derived from the action principle, implies that field theories are microscopically reversible by design: differential changes experience no significant obstacles in a system and may be trivially undone. Yet, when summed macroscopically, in the context of an environment, such individually reversible changes lead to the well known irreversible behaviours of thermodynamics: the reversal of paths through an environmental landscape would require the full history of the route taken. Classical field theory thus forms a basis for both the microscopic and the macroscopic. When applied to quantum mechanics, the classical framework is sometimes called the first quantization . The first quantization may be considered the first stage of a more complete theory, which goes on to deal with the issues of many–particle symmetries and interacting fields. Quantum mechanics is classical field theory with additional assumptions about measurement. The term quantum mechanics is used as a name for the specific theory of the Schrödinger equation, which one learns about in undergraduate studies, but it is also sometimes used for any fundamental description of physics, which employs the measurement axioms of Schrödinger quantum mechanics, i.e. where change is expressed in terms of fields and groups.
In contemporary field theory, the word classical is reserved for an analytical framework in which the local equations of motion provide a complete description of the evolution of the fields. Classical field theory is a differential expression of change in functions of space and time, which summarizes the state of a physical system entirely in terms of smooth fields. The differential (holonomic) structure of field theory, derived from the action principle, implies that field theories are microscopically reversible by design: differential changes experience no significant obstacles in a system and may be trivially undone. Yet, when summed macroscopically, in the context of an environment, such individually reversible changes lead to the well known irreversible behaviours of thermodynamics: the reversal of paths through an environmental landscape would require the full history of the route taken. Classical field theory thus forms a basis for both the microscopic and the macroscopic. When applied to quantum mechanics, the classical framework is sometimes called the first quantization . The first quantization may be considered the first stage of a more complete theory, which goes on to deal with the issues of many–particle symmetries and interacting fields. Quantum mechanics is classical field theory with additional assumptions about measurement. The term quantum mechanics is used as a name for the specific theory of the Schrödinger equation, which one learns about in undergraduate studies, but it is also sometimes used for any fundamental description of physics, which employs the measurement axioms of Schrödinger quantum mechanics, i.e. where change is expressed in terms of fields and groups.
Nothing can be made out of nothing. William Shakespeare , King Lear Historically, quantization of the motion of particles was developed first. The state was described by a wave function and observables by operators. When dealing with interactions between particles and fields, such as the electromagnetic field, the fields were treated classically. Classical field equations look like the quantum mechanical equations for the wave function of the field quanta. For example, the Klein–Gordon classical field equation is similar to the quantum mechanical wave equation for a relativistic spinless particle. Quantizing the fields, leading to quantum field theory, appears to be quantizing a theory that has already been quantized; hence the name “second quantization.” In reality, there is only one quantization and one quantum theory. The method of second quantization is important in the study of many-particle systems. It enables us to express many-body operators in terms of creation and annihilation operators, thus rendering calculations less cumbersome. Moreover, the method makes it possible to treat systems with a variable number of particles; that is why the method initially emerged in the context of quantum field theory. In Chapter 1 we indicated that any one-particle wave function may be expanded in a complete set of states. In this chapter, we show that products of single-particle states, when properly symmetrized, form an orthonormal basis for the expansion of the wave function of an N -particle system.
In the standard formulation of quantum mechanics, observables are represented by operators in a Hilbert space and states by vectors in it. Quantum field theory too is formulated in terms of field operators. States in field theory include the vacuum state, one particle states, two particle states and so on. One does not measure operators or state vectors, however; measurements involve eigenvalues of operators or transition probabilities between states. It turns out that one can formulate quantum mechanics and field theory without using operators. Indeed, the probability amplitude for a transition from a position q1 at time t1 to a position q2 at time t2 can be expressed as a path integral ∫ dt L(q(t),q˙(t)), (5.1) where q(t) runs over all possible functions satisfying the conditions q(t1) = q1, q(t2) = q2 (5.2) and the integral over q(t) is an integral over all such paths of the exponential evaluated for each path. The measure of integration is not well defined a priori, but it is possible to give prescriptions that make sense. In classical mechanics, positions at two different times determine a path from the second order differential equation of motion, which yields a definite solution if two conditions are imposed. In contrast, quantum mechanics does not allow us to talk about trajectories. The path integral formulation manifestly refers to the myriad paths that may be taken by a particle in going from one point to another. It goes so far as to assign a probability amplitude to each path, and it is related to the classical action corresponding to that path. However, it is complex and it also involves a superposition, characteristic of quantum mechanics.
We propose a new approach concerning the introduction of time-irreversibility in statistical mechanics. It is based on a transition function defined in terms of path integral and verifying a time-irreversible equation. We show first how dynamic processes may enter in the description of equilibrium states. In order to do that a characteristic time is associated with closed paths. For large isolated systems at equilibrium or for systems in contact with a thermostat our results are identical with those obtained with the Gibbs ensemble methods. For a model used in the microscopic approaches of the brownian motion no new basic assumption is required to predict a transition from a quantum state to a classical one exhibiting a time-irreversible behavior. This demonstration is sufficient to show that very well accepted approximations can lead to time-irreversible behaviors for a large class of systems. The difference between our work and the system+reservoir approaches is underlined. Here equilibrium states and irreversible processes are described on the same footing representing a progress in the question of time-irreversibility in statistical physics. The transition function is also used for describing a small system for which there is no thermodynamics. By adding to the transition function a second one characterizing the reverse motion we may describe time-reversible systems. In a simple case we replace two real valued transition functions by a complex function verifying a Schrodinger like equation. From this we see how to break the time-reversibility of this equation and how to investigate the connection quantum mechanics-thermodynamics from a very fundamental point of view.
Quantum field theory is the ultimate expression of quantum mechanics. Its formalism provides the framework for all modern fundamental theories about the nature of matter. However, it involves difficult and intricate mathematics, which is why physics students don’t generally come across it until postgraduate level, and then only if their subject specialism requires the insights that it provides. In ordinary quantum theory, observable properties are represented by Hermitian operators, with eigenstates that form a basis for the expansion of a system’s states. Philosophers have been debating the correctness of such arguments, but now it seems that experimental science has come along and shown that, at the quantum level at least, the objects we study have no substance to them, independent of their properties. In classical physics, interactions between particles can be considered by either an examination of forces or from the potential energies involved.
This manuscript introduces the Transformation Capacity Framework (TCF), an axiomatic proposal deriving special relativity, quantum mechanics, and general relativity from a single discrete substrate. Rather than quantizing classical spacetime or embedding quantum fields onto a pre-existing geometric background, we model the physical universe as a locally finite, reversible causal graph governed by a conserved transformation capacity budget. Crucially, within this architecture, macroscopic time is not a fundamental background dimension; rather, it emerges purely as a thermodynamic receipt, a measurable accumulation of discrete, microscopic physical transformations. We establish that a conserved variance partition between internal state updates and external spatial diffusion uniquely generates a macroscopic Lorentzian metric in the continuum limit, resolving the Minkowski signature without prior geometric assumptions. Because time is the accumulated length of change, relativistic time dilation arises mechanically as a capacity trade-off: external spatial routing exhausts the localized capacity available for internal state updates. Primitive operation symmetry implies universal Hamiltonian scaling, recovering the equivalence principle. Microscopic information preservation forces the emergence of complex Hilbert space structure, unitary time evolution via Stone's theorem, and the Born rule via Gleason's theorem. Finally, the macroscopic thermodynamic limit of the network yields the Einstein field equations as the unique low-energy effective theory, where mathematical singularities are resolved via finite queue saturation. This framework demonstrates that spacetime geometry and quantum mechanics are complementary macroscopic manifestations of a resource-constrained, information-preserving computational network.
The classical system is usually quantized using the canonical formalism, i.e., in the language of coordinates and momenta. In addition to this “purely algebraic” ambiguity, there is also an ambiguity in the choice of realization of the operators p^ and q^: the commutation relations do not determine it uniquely, but up to a unitary equivalence (the well known uniqueness theorem of von Neumann pertains only to the Weyl form of the commutation relations). The realization must be chosen on the basis of additional considerations regarding the nature of the canonical variables p and q. In this book the term “theory” is used in various ways: in a broad sense (quantum field theory, relativistic theory) and in a narrow sense, to refer to a particular system or class of systems with certain dynamics. In quantum theory there are also systems whose operators corresponding to canonical variables satisfy not commutation relations, but anticommutation relations.