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This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.
In particular, this means that a system of N interacting particles in 3 dimensions is described by one vector whose components in a basis where all the X are diagonal is a mathematical function of 3N-dimensional space describing all their possible positions, effectively a much bigger collection of values than the mere collection of N three ...
The construction of the Standard Model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.
The term "particle" in this context refers to gaseous particles only (atoms or molecules), and the system of particles is assumed to have reached thermodynamic equilibrium. [1] The energies of such particles follow what is known as Maxwell–Boltzmann statistics , and the statistical distribution of speeds is derived by equating particle ...
Instantons are used in nonperturbative calculations of tunneling rates. Instantons have properties similar to particles, specific examples include: Calorons, finite temperature generalization of instantons. Merons, a field configuration which is a non-self-dual solution of the Yang–Mills field equation. The instanton is believed to be ...
For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system of N particles, all of these equations apply to each particle in the system). The equation of motion for a particle of constant mass m is Newton's second law of 1687, in modern vector notation F = m a , {\displaystyle \mathbf {F} =m ...
If the particles have the same physical properties, the n j s run over the same range of values. Let ε(n) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies. The partition function of the system is
[1] [2] [3] In classical mechanics for instance, in the action formulation, extremal solutions to the variational principle are on shell and the Euler–Lagrange equations give the on-shell equations. Noether's theorem regarding differentiable symmetries of physical action and conservation laws is another on-shell theorem.