What are the units of a scalar field if I only impose $c=1$?
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hey mate
good question
answer:-
Whenever two things are to be added together, one typically needs to check whether this actually makes sense, and an addition is said to make sense, in principle, when the units match up.
Yet, matching units clearly aren't enough:
Both action (like ℏ) and angular momentum Lshare the SI-units Js.Though adding such quantities can quickly be dismissed by stating that →L is a directed quantity and vectors and scalars can't (usually) be added together either.Setting c=1, as commonly done in relativistic mechanics, various units become one and the same:[length]=[time], [mass]=[energy]=[momentum], [acceleration]=[frequency] etc.those do mostly make sense with 4-vectors, recognizing that, say, time is orthogonal to space, although I haven't seen [acceleration]=[frequency] used before.going a step further, introducing Planck units derived from natural constants, all units become powers of [length]: (done by hand, errors possible)[1]=[#particles]=[%]=[odds]=[velocity]=[entropy]=[action]=[angular momentum]=[charge]=[resistance][length]=[time]=[inductance][length−1]=[energy]=[mass]=[momentum]=[frequency]=[acceleration]=[torque]=[capacitance]=[current]=[voltage]=[temperature]=[chemical potential][length−2]=[force]=[magnetic flux density][length−4]=[pressure]probably many many more (which are actually in use somewhere - in principle there would of course be infinitely many such relations)
So for many of those I can clearly see why they couldn't possibly be added together in a consistent manner:
Some of these quantities are fundamentally scalars, others are axial vectors and yet other ones are polar vectors. - in normal vector calculus, those three can't just be added together.
Particle count and relative quantities are also different in that they are defined on different domains; respectively N and [0,1]. All the other above quantities are defined either on R or on [0,∞[, although experimental data and theories that conform to them may limit some of them, like electric charge, further.
So the necessary conditions for a properly defined addition I found thus far are:
matching unitsmatching domainsmatching dimensionmatching transformational behavior or covariance (? - I am not well versed enough in GR to know that this always is a problem. I am vaguely aware of co- and contra-variant tensors and such, but I can't recall whether a co- and a contra-variant tensor from the same space can usually be added together. What I do not count, though, is addition as it might happen in Geometric Algebra, where you can have multivectors. In that context, I'm specifically asking about addition of like blades. In that case the fact that axial and polar vectors can't be added together becomes the fact that they correspond to vectors or bivectors.)
hey mate
good question
answer:-
Whenever two things are to be added together, one typically needs to check whether this actually makes sense, and an addition is said to make sense, in principle, when the units match up.
Yet, matching units clearly aren't enough:
Both action (like ℏ) and angular momentum Lshare the SI-units Js.Though adding such quantities can quickly be dismissed by stating that →L is a directed quantity and vectors and scalars can't (usually) be added together either.Setting c=1, as commonly done in relativistic mechanics, various units become one and the same:[length]=[time], [mass]=[energy]=[momentum], [acceleration]=[frequency] etc.those do mostly make sense with 4-vectors, recognizing that, say, time is orthogonal to space, although I haven't seen [acceleration]=[frequency] used before.going a step further, introducing Planck units derived from natural constants, all units become powers of [length]: (done by hand, errors possible)[1]=[#particles]=[%]=[odds]=[velocity]=[entropy]=[action]=[angular momentum]=[charge]=[resistance][length]=[time]=[inductance][length−1]=[energy]=[mass]=[momentum]=[frequency]=[acceleration]=[torque]=[capacitance]=[current]=[voltage]=[temperature]=[chemical potential][length−2]=[force]=[magnetic flux density][length−4]=[pressure]probably many many more (which are actually in use somewhere - in principle there would of course be infinitely many such relations)
So for many of those I can clearly see why they couldn't possibly be added together in a consistent manner:
Some of these quantities are fundamentally scalars, others are axial vectors and yet other ones are polar vectors. - in normal vector calculus, those three can't just be added together.
Particle count and relative quantities are also different in that they are defined on different domains; respectively N and [0,1]. All the other above quantities are defined either on R or on [0,∞[, although experimental data and theories that conform to them may limit some of them, like electric charge, further.
So the necessary conditions for a properly defined addition I found thus far are:
matching unitsmatching domainsmatching dimensionmatching transformational behavior or covariance (? - I am not well versed enough in GR to know that this always is a problem. I am vaguely aware of co- and contra-variant tensors and such, but I can't recall whether a co- and a contra-variant tensor from the same space can usually be added together. What I do not count, though, is addition as it might happen in Geometric Algebra, where you can have multivectors. In that context, I'm specifically asking about addition of like blades. In that case the fact that axial and polar vectors can't be added together becomes the fact that they correspond to vectors or bivectors.)
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