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# Dot Product watch

1. Does anyone know how the 3-vector dot product is defined in einstein notation in a Minkowski space with metric (+1, -1, -1, -1). Is it or ? I always thought it was defined as the latter, but I was just working through a problem on writing the EM Lagrangian in terms of and in the solution it had which seems to suggest otherwise.
Any help would be greatly appreciated,
Thanks
2. Euh?

In Minkowski space with (1,-1,-1,-1) you have . Or if you prefer, you can write where is one-dimensional and is three-dimensional, and then (with summation convention).
3. On the attached file, on the second page Q2 part ii, the final line.
I don't really understand what's going on because we working with so the summations have to be over upper and lower indices, but then in the last bit when we've changed to the 3 dimensional vectors the summation is just over upper indices and I don't really understand how this works. I would have thought the 3 dimensional dot product would still be because spatial indices pick up a - sign when switching between upper and lower indices. I'm a little confused by this.
Attached Images
4. QFTsol2.pdf (693.4 KB, 61 views)
5. Possibly I'm being dim, but I can't actually work out which line you're referring to. Maybe you could LaTeX it?
6. (Original post by DFranklin)
Possibly I'm being dim, but I can't actually work out which line you're referring to. Maybe you could LaTeX it?
It's the line that says with a remark noting the same level indices.
7. That's the last line of (iii), not (ii), surely?

And I confess I don't entirely understand the problem. You say you'd expect , but surely that's exactly what they've done?

[I should say that this I never really covered co/contravariant tensors in any kind of detail, so this is a bit beyond my competence, but it looks OK to me].
8. (Original post by DFranklin)
That's the last line of (iii), not (ii), surely?

And I confess I don't entirely understand the problem. You say you'd expect , but surely that's exactly what they've done?

[I should say that this I never really covered co/contravariant tensors in any kind of detail, so this is a bit beyond my competence, but it looks OK to me].
Woops, it was part iii.

Don't they have which is the negative of what I expected.

(I haven't covered co/contravariant tensors except in terms of defining one from the other with the metric. I have no idea what the difference between them is, but I don't think I need to know that anyway.)
9. I don't understand. Again, you say you expect , so why are you surprised that ?

Can you narrow things down to one equation (i.e. "X = Y" rather than "X = Y = Z = W") so it's clear exactly what you don't think follows?
10. (Original post by DFranklin)
I don't understand. Again, you say you expect , so why are you surprised that ?

Can you narrow things down to one equation (i.e. "X = Y" rather than "X = Y = Z = W") so it's clear exactly what you don't think follows?
I'm happy with
What I don't fully understand is what refers to. I'd always thought that but they seem to have defined
11. OK; that's the only point I wasn't sure about, but then it's all about what you mean by E^2, not really anything about summation convention and/or raising/lowering tensors etc.

I took it to mean simply (i.e. the square of the size of E in a "normal" metric), but it seems a valid thing to query with your lecturer.

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