Dimensions of P and ##\omega##

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Discussion Overview

The discussion revolves around the dimensions of momentum eigenvalues and angular frequency (##\omega##) in the context of polymer quantum mechanics. Participants explore the implications of using natural units, particularly in relation to measurements in GeV and the dimensional analysis required for making equations dimensionless.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the dimensions of momentum eigenvalues, suggesting they have dimensions of ##(mass)^{-1}## and seeks clarification on how ##\omega## can have dimensions of mass.
  • Another participant proposes that the conventions in polymer physics may align with those in high-energy physics (HEP), where natural units are used, implying that only one dimension remains, typically chosen as energy or length.
  • A participant seeks to understand the implications of measuring mass in GeV, questioning whether this leads to momentum being treated as having dimensions of ##(mass)^{-1}##.
  • There is a request for clarification on who claims the dimensions mentioned, indicating a need for specific references.
  • Some participants clarify that in natural units, the units for mass, energy, and momentum are equivalent, all measured in GeV, which challenges the initial assumption about the dimensions of momentum.
  • Further clarification is provided that in natural units, angular momenta and actions are dimensionless, and if ##k_{\text{B}}## is set to 1, temperatures are also measured in GeV.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the dimensional analysis of momentum and its relationship to mass in GeV. There is no consensus on the interpretation of these dimensions, and multiple viewpoints are presented without resolution.

Contextual Notes

Participants reference the use of natural units and the implications for dimensional analysis, but the discussion does not resolve the assumptions or definitions that may affect the interpretation of these dimensions.

AHSAN MUJTABA
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TL;DR
Dimensional analysis in quantum mechanics, of physical quantities.
I am studying polymer quantum mechanics. In it, they say that the momentum, ##p## eigenvalue, has the dimensions of ##(mass)^{-1}## and similarly ##\omega## has the dimensions of ##mass##. How it is possible, please someone explain that to me. Even a little hint would work.
I don't get it. Also, I would require some assistance regarding the units of Planck's reduced mass, ##M_{PI}^{2}##. How can it be measured in terms of GeV?
 
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I'm not sure about the conventions in the polymer-physics community, but from what you describe, I guess they use a similar convention as we do in HEP physics, i.e., they use natural units, setting ##\hbar=c=1##. In such a system of units you have only one dimension left. You can choose energies (MeV or GeV) or lengths (usually fm). Usually one uses GeV for masses, energies, momenta, frequencies, wave numbers etc. and fm for lengths and times. All you have to keep in mind to convert from GeV to 1/fm or from fm to 1/GeV is that ##\hbar c \simeq 0.197 \; \text{GeV} \; \text{fm}##.
 
Does that imply that the dimension of momentum eigenvalue becomes inverse of mass? You say that we measure mass in terms of GeV. So, due to that unit(GeV), does the dimension of momentum become (mass)##^{-1}##?
 
I need to understand these dimensions as I am making some equations dimensionless for my tasks.
 
AHSAN MUJTABA said:
they say
Who says? Can you give a specific reference?
 
AHSAN MUJTABA said:
Does that imply that the dimension of momentum eigenvalue becomes inverse of mass? You say that we measure mass in terms of GeV. So, due to that unit(GeV), does the dimension of momentum become (mass)##^{-1}##?
Not in the "natural" units @vanhees71 described. In those units, as he said in post #2, the unit of momentum is the same as the unit of mass, energy, etc.--all are measured in a unit like GeV.
 
AHSAN MUJTABA said:
Does that imply that the dimension of momentum eigenvalue becomes inverse of mass? You say that we measure mass in terms of GeV. So, due to that unit(GeV), does the dimension of momentum become (mass)##^{-1}##?
In the natural system of units, where ##\hbar=c=1## the unit for mass, energy, and momentum is GeV (or any other energy unit you prefer). Lengths and times are usually measured in fm. Angular momenta and actions are dimensionless.

If you also set ##k_{\text{B}}=1## then also temperatures are measured in GeV.
 

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