Editing Dimensional analysis (section)

== Dimensionless concepts ==

=== Constants ===
{{Main|Dimensionless quantity}}
The dimensionless constants that arise in the results obtained, such as the {{math|''C''}} in the Poiseuille's Law problem and the {{math|''κ''}} in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation.  Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity.  This observation can allow one to sometimes make "[[back of the envelope]]" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.

=== Formalisms ===
Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the [[Ising model]] can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, {{math|''χ''}}) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be {{math|~ 1/''χ''{{sup|''d''}}}}, where {{math|''d''}} is the dimension of the lattice.

It has been argued by some physicists, e.g., [[Michael Duff (physicist)|Michael J. Duff]],<ref name="duff" /><ref>{{cite arXiv |last=Duff |first=Michael James |eprint=hep-th/0208093v3 |title=Comment on time-variation of fundamental constants |date=July 2004}}</ref> that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: [[Speed of light|{{math|''c''}}]], [[Planck constant|{{math|''ħ''}}]], and [[Gravitational constant|{{math|''G''}}]], in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.

Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants {{math|''ħ''}}, {{math|''c''}}, and {{math|''G''}} (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit {{math|''c'' → ∞}}, {{math|''ħ'' → 0}} and {{math|''G'' → 0}}. In problems involving a gravitational field the latter limit should be taken such that the field stays finite.