COORDINATE TRANSFORMATIONS IN CLASSICAL FIELD THEORY

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1 COORDINATE TRANSFORMATIONS IN CLASSICAL FIELD THEORY Link to: physicspages home page. To leave a comment or report an error, please use the auiliary blog. Reference: W. Greiner & J. Reinhardt, Field Quantization, Springer- Verlag 996), Chapter 2, Section 2.4. The various conservation laws of physics energy, linear and angular momentum) can be derived from the invariance of a system under coordinate transformations. To prepare for Noether s theorem, which is a general theorem allowing us to derive these conservation laws, we need to consider how the fields themselves transform under coordinate transformations. In what follows, we ll consider only infinitesimal transformations, and we define a general transformation as µ µ + δ µ ) Note that µ and µ both refer to the same physical point in space; they simply represent two different coordinate systems referring to this same point. Under this transformation, the mathematical function describing the field will change as well, so we can write φ r ) φ r ) + δφ r ) 2) where the subscript r labels which field we re talking about. Again, φ r ) and φ r ) both represent the same field at the same point in space-time; they are just epressed in different coordinate systems. At this point, it s useful to have a look at a specific eample. Suppose the field φ is a vector field in two dimensions we ll drop the r subscript, as we re dealing with only one field). We ll see what happens if we rotate the coordinate system through an angle θ, as in the diagram, where the unprimed system is drawn in black and the primed system in blueo.

2 COORDINATE TRANSFORMATIONS IN CLASSICAL FIELD THEORY 2 In the unprimed system, φ consists of horizontal vectors with a magnitude equal to thei 2 coordinate. φ) 2 Under a rotation, the coordinates transform according to 3) 2 cosθ sinθ sinθ cosθ Inverting the rotation gives 2 cosθ + 2 sinθ sinθ + 2 cosθ 4) cosθ sinθ 2 sinθ cosθ 2 cosθ 2 sinθ sinθ + 2 cosθ For our eample vector field 3, we have, by inserting 3 into 4 that is, in 4 we set 2 and 2 ): φ ) 2 cosθ sinθ cosθ + 2 cos2 θ 2 sinθ sin2 θ 6) 2 sinθ cosθ As we can see from the diagram by looking at the magenta vector, the vector in the unprimed system is parallel to the ais, with length 2 as given by 3. If we rotate the coordinate aes by the angle θ we get the primed system shown as the blue aes, and we can see that in that system, the magenta vector has a positive component in the direction and a negative component in the 2 direction. However, the length of the vector remains the same in both systems, since the vector itself doesn t change when we 5)

3 COORDINATE TRANSFORMATIONS IN CLASSICAL FIELD THEORY 3 simply rotate the coordinates. We ll eplain the green vector later on in this post.) Since we ll deal primarily with infinitesimal transformations from now on, we ll do the rest of the analysis using that approimation. For the rotation eample above, if θ is now an infinitesimal angle I suppose I should write it as δθ but this just clutters up the notation, so just remember that θ is infinitesimal and all will be well.), then we have, to first order in θ, cosθ and sinθ θ, so for a general rotation + 2 θ 2 θ + 2 δ 2 θ θ For the specific eample above, to first order in θ φ ) 2 θ θ 2 θ Plugging 3 and 9 into 2, we get δφ) φ ) φ) 2 θ 7) 8) 9) ) Up to now, we ve considered what happens at one specific point when the coordinate system is varied. The variation δφ) is the result of varying both the coordinate system and the effect this variation has on the form of the field epression. In practice, another kind of variation, called the modified or total variation is defined by δφ r ) φ r ) φ r ) ) Note that the difference between δφ r ) and δφ r ) is that the φ r term is evaluated at in the former and at in the latter. This notation is somewhat confusing, since in 2, both and refer to the same point in the plane, while in the latter, the in φ r ) is a different point from the in φ r ). We can illustrate this by looking again at the above diagram. The point in the unprimed system is at around, 2 ),2) it s the location of the tail of the magenta vector, identified by the dotted black lines). The notation φ r ) means that we insert the same numerical values for, 2 ) into the function φ r, that is, we set, 2 ),2). This gives the location indicated by the tail of the green vector, as identified by the dotted blue lines. Since this location is higher up the 2 ais than the magenta vector, the green vector is longer than the magenta vector, so that φ r ) and φ r )

4 COORDINATE TRANSFORMATIONS IN CLASSICAL FIELD THEORY 4 now refer to two different vectors. The quantity δφ r ) therefore measures the change in the field due solely to the transformation of the coordinates. We can, nevertheless, derive a relation between δφ r ) and δφ r ). Starting from, we have δφ r ) φ r ) φ r ) 2) φ r ) φ r ) + φ r ) φ r ) 3) φ r ) φ r ) ) + δφ r ) 4) δφ r ) φ r ) µ δ µ 5) δφ r ) φ r ) µ δ µ 6) In the penultimate line, we replaced φ r ) φ r ) by its first order term in the Taylor epansion, and in the last line, we approimated φ r ) by φ r ), again valid to first order. As an eample, we can apply this formula to the above vector field. Starting with, we have, using 9 and 3 δφ) φ ) φ) 7) θ ) 2 θ θ 9) 2 θ Now we can check 6. From 3 we have φ) φ) 2 2) 2) From 8, we have

5 COORDINATE TRANSFORMATIONS IN CLASSICAL FIELD THEORY 5 φ r ) µ δ µ Combining this with we get δ + θ + θ δ 2 22) 23) 24) δφ r ) δφ r ) φ r ) δ µ 25) µ θ + 26) 2 θ θ 27) 2 θ which agrees with 9. Finally, we can note a couple of formulas concerning the derivative of the two variations δφ r ) and δφ r ). Since δφ r ) depends only on and not on ), the derivative commutes with the variation: ) φr ) δφ r ) δ µ µ 28) The other variation δφ r ) is a bit trickier, since it involves as well as. However, using the chain rule, we can find its derivative. I ll use the shorthand µ / µ and µ / µ. µ δφ r )) µ φ ) µ φ r ) 29) µφ ) µ φ r ) + µ φ ) µφ ) 3) δ µ φ r )) + νφ )) µ ν) µφ ) 3) We can now use on the middle term: Combining the last two terms, we get µ ν µ ν + δ ν ) 32) δ µν + µ δ ν 33)

6 COORDINATE TRANSFORMATIONS IN CLASSICAL FIELD THEORY 6 ν φ )) δ µν + µ δ ν ) µφ ) νφ )) µ δ ν 34) ν φ r )) µ δ ν 35) Again, the last step is valid to first order in the variations. Thus we have µ δφ r )) δ µ φ r )) + ν φ r )) µ δ ν 36) PINGBACKS Pingback: Noether s theorem and conservation laws