''For classical dynamics at relativistic speeds, see

relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...

.'' Relativistic dynamics refers to a combination of relativistic and

quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...

concepts to describe the relationships between the motion and properties of a relativistic system and the forces acting on the system. What distinguishes relativistic dynamics from other physical theories is the use of an invariant scalar evolution parameter to monitor the historical evolution of

space-time In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...

events. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved. Twentieth century experiments showed that the physical description of

microscopic The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens (optics), lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded a ...

and submicroscopic objects moving at or near the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

raised questions about such fundamental concepts as space, time, mass, and energy. The theoretical description of the physical phenomena required the integration of concepts from relativity and

quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...

Vladimir Fock Vladimir Aleksandrovich Fock (or Fok; russian: Влади́мир Алекса́ндрович Фок) (December 22, 1898 – December 27, 1974) was a Soviet Union, Soviet physicist, who did foundational work on quantum mechanics and quantum ...

was the first to propose an evolution parameter theory for describing relativistic quantum phenomena, but the evolution parameter theory introduced by

Ernst Stueckelberg Ernst Carl Gerlach Stueckelberg (baptised as Johann Melchior Ernst Karl Gerlach Stückelberg, full name after 1911: Baron Ernst Carl Gerlach Stueckelberg von Breidenbach zu Breidenstein und Melsbach; 1 February 1905 – 4 September 1984) was a S ...

is more closely aligned with recent work. Evolution parameter theories were used by

Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...

, Schwinger and others to formulate quantum field theory in the late 1940s and early 1950s.

Silvan S. Schweber Silvan Samuel Schweber (10 April 1928 in Strasbourg – 14 May 2017) was a French-born American theoretical physicist and historian of science. Biography In 1944 Schweber began to study chemistry at the City College of New York and in 1947 moved t ...

wrote a nice historical exposition of Feynman’s investigation of such a theory. A resurgence of interest in evolution parameter theories began in the 1970s with the work of Horwitz and Piron, and Fanchi and Collins.

Invariant Evolution Parameter Concept

Some researchers view the evolution parameter as a mathematical artifact while others view the parameter as a physically measurable quantity. To understand the role of an evolution parameter and the fundamental difference between the standard theory and evolution parameter theories, it is necessary to review the concept of time. Time t played the role of a monotonically increasing evolution parameter in classical Newtonian mechanics, as in the force law F = dP/dt for a non-relativistic, classical object with momentum P. To Newton, time was an “arrow” that parameterized the direction of evolution of a system.

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

rejected the Newtonian concept and identified t as the fourth coordinate of a space-time four-

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

. Einstein's view of time requires a physical equivalence between coordinate time and coordinate space. In this view, time should be a reversible coordinate in the same manner as space. Particles moving backward in time are often used to display antiparticles in Feynman-diagrams, but they are not thought of as really moving backward in time usually it is done to simplify notation. However a lot of people think they are really moving backward in time and take it as evidence for time reversibility. The development of non-relativistic

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

in the early twentieth century preserved the Newtonian concept of time in the Schrödinger equation. The ability of non-relativistic quantum mechanics and special relativity to successfully describe observations motivated efforts to extend quantum concepts to the relativistic domain. Physicists had to decide what role time should play in relativistic quantum theory. The role of time was a key difference between Einsteinian and Newtonian views of classical theory. Two hypotheses that were consistent with

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...

were possible:

Hypothesis I

Assume t = Einsteinian time and reject Newtonian time.

Hypothesis II

Introduce two temporal variables: *A coordinate time in the sense of Einstein *An invariant evolution parameter in the sense of Newton ''Hypothesis I'' led to a relativistic probability conservation equation that is essentially a re-statement of the non-relativistic continuity equation. Time in the relativistic probability conservation equation is Einstein’s time and is a consequence of implicitly adopting ''Hypothesis I''. By adopting ''Hypothesis I'', the standard paradigm has at its foundation a temporal paradox: motion relative to a single temporal variable must be reversible even though the second law of

thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...

establishes an “arrow of time” for evolving systems, including relativistic systems. Thus, even though Einstein’s time is reversible in the standard theory, the evolution of a system is not time reversal invariant. From the perspective of ''Hypothesis I'', time must be both an irreversible arrow tied to entropy and a reversible coordinate in the Einsteinian sense. The development of relativistic dynamics is motivated in part by the concern that ''Hypothesis I'' was too restrictive. The problems associated with the standard formulation of relativistic quantum mechanics provide a clue to the validity of ''Hypothesis I''. These problems included negative probabilities, hole theory, the Klein paradox, non-covariant expectation values, and so forth. Most of these problems were never solved; they were avoided when

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

(QFT) was adopted as the standard paradigm. The QFT perspective, particularly its formulation by Schwinger, is a subset of the more general Relativistic Dynamics. Relativistic Dynamics is based on ''Hypothesis II'' and employs two temporal variables: a coordinate time, and an evolution parameter. The evolution parameter, or parameterized time, may be viewed as a physically measurable quantity, and a procedure has been presented for designing evolution parameter clocks.Fanchi, J.R. (1993): Parametrized Relativistic Quantum Theory (Kluwer, Dordrecht) By recognizing the existence of a distinct parameterized time and a distinct coordinate time, the conflict between a universal direction of time and a time that may proceed as readily from future to past as from past to future is resolved. The distinction between parameterized time and coordinate time removes ambiguities in the properties associated with the two temporal concepts in Relativistic Dynamics.

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