Stefan Ram

2023-06-24 20:37:05 UTC

To better understand symmetries, I have a few questions, but for

simplicity's sake I'll start with one:

Let a hypothetical one-dimensional world consist of a ray with

values x>=0. This world is completely empty except for a mass

point with unit mass 1 at x=1. This is described by a "mass

density" R(x), which is zero everywhere except for R(1)=1.

Now a scientist comes along and says: I formally extend this world

to a two-dimensional world with the coordinates (x,y). The mass

density R(x,y):=R(x) is everywhere equal to 0, except for a mass in

the form of a straight line {(1,y)|yeR}. This world is invariant with

respect to y. A translation y'=y+y0 results in the same world again.

So, there is a preserved quantity, which I call the "y-momentum".

Now, there are two reactions: One praises the "deep result". Others

say that y is just a "redundant, unphysical coordinate" that has

no meaning at all, and that the result is completely irrelevant.

So, is the y-invariance of the two-dimensional world irrelevant

or meaningful? Why?

[[Mod. note -- In order to have unit mass, doesn't your mass density

need to be a Dirac delta-function? -- jt]]

simplicity's sake I'll start with one:

Let a hypothetical one-dimensional world consist of a ray with

values x>=0. This world is completely empty except for a mass

point with unit mass 1 at x=1. This is described by a "mass

density" R(x), which is zero everywhere except for R(1)=1.

Now a scientist comes along and says: I formally extend this world

to a two-dimensional world with the coordinates (x,y). The mass

density R(x,y):=R(x) is everywhere equal to 0, except for a mass in

the form of a straight line {(1,y)|yeR}. This world is invariant with

respect to y. A translation y'=y+y0 results in the same world again.

So, there is a preserved quantity, which I call the "y-momentum".

Now, there are two reactions: One praises the "deep result". Others

say that y is just a "redundant, unphysical coordinate" that has

no meaning at all, and that the result is completely irrelevant.

So, is the y-invariance of the two-dimensional world irrelevant

or meaningful? Why?

[[Mod. note -- In order to have unit mass, doesn't your mass density

need to be a Dirac delta-function? -- jt]]