The relationship between difference and ratio and a proposal: Equivalence of temperature and time, and the first spontaneous symmetry breaking

In mathematics the concept of 'a group' is used to formulate, in a most general way, any kind of mathematical operation, for example, addition, multiplication or rotation. Generally, a mathematical group consists of 'elements' a, b, c, ... and an instruction that attributes to each pair of elements, say a and b, a new element a ° b in such a way that three group axioms G1, G2 and G3 strictly apply. Abelian groups are commutative for which group axiom G4 applies as well:The mathematical operation of addition in conjunction with elements that are integers, {a, b, c, ...} ∈ ？, is an abelian group. To specify, replace the above general formalism with a specific one, i.e., replace ° with +, e with 0, and a–1 with –a. The mathematical operation of multiplication in conjunction with elements that are positive real numbers, {a, b, c, ...} ∈ ？+, is also an abelian group: ° : = ？ , e : = 1 , and a ？ 1 : = 1 / a . Note that, whereas the addition of real numbers is an abelian group, axiom G3 usually does not apply to the multiplication of positive integers {a, b, c, ...} ∈ ？+ because generally {a–1, b–1, c–1, ...} ？ ？. Most commonly, mathematical groups are used to describe geometrical symmetries; the vast majority of mathematical groups are non-abelian (axiom G4 does not apply), in particular, when the parameter space is higher than 2D (two-dimensional).In physics and physical cosmology this formalism is extremely useful for the description of spontaneous symmetry breaking into different fundamental forces through the application of gauge theories, such as in Quantum Electrodynamics (QED) or Quantum Chromodynamics (QCD) being the basis of the Standard Model. A gauge theory is a type of field theory in which the Lagrangian is invariant under a certain continuous group of local transformations. The Lagrangian of a dynamical system is a function that summarises the dynamics of the system. Common to all these theories is the utilization of Legendre transformations. The