Some questions: 1. At 8:21, shouldn't there be a factor of 1/2? since the inner product is expanded into the sum of 2 terms 2. At 9:01, can the operator δ be treated as a constant with respect to the basis |ψ>? (since it escapes the bra-ket) 3. At 11:11, it seems like the LHS of the two equations stay the same but the RHS of the second equation got exponentiated? and I might be mistaken here, but shouldn't an integration remove all of the "δ" in the equation, including |ψ+δψ>?
1) I see what you mean. The way to interpret the action operator inside that inner product is this: the operator acts like a differential operator (on the right state, but also on the left state). Let me remind that this is a "heuristic" derivation, so the notation is not always rigorous here. Any books on QFT that deal with the Schwinger formalism should cover this more rigorously. 2) "delta" represents the variation of the state due to a small change of the field. In general, the state changes if we perturb the field, so I would not consider your interpretation correct. "delta" can act both on the "bra" and on the "ket". 3) yes, at 11:11 we have a simple exponentiation. The variable of integration on the left is the entire inner product between the "bra" and the "ket" you see in the expression. The result should be the logarithm as written in the video (the delta in the inner product does not disappear)
Another way to think about your first question. Consider the evolution of the change in the action between two states A and B. We can write the inner product: < B | δS | A >. Based upon the considerations in the video, δS should depend only on the final and initial states, so it could be written as: δS= S_B - S_A (where S_B is an operator that acts only on state B, and S_A acts only on state A). Let's assume C is another state (that you can think of as being "in between" A and B). You can write: δS = S_B - S_A = ( S_B - S_C ) + ( S_C-S_A ) Therefore: < B | δS | A > = < B | ( S_B - S_C ) | A > + < B | ( S_C - S_A ) | A > Now, those differences in the action are going to create some small perturbations. In particular, S_B - S_C will act on < B |, whereas ( S_C - S_A ) will act on | A >. This serves as another "justification" of the formula you were confused about.
Some questions:
1. At 8:21, shouldn't there be a factor of 1/2? since the inner product is expanded into the sum of 2 terms
2. At 9:01, can the operator δ be treated as a constant with respect to the basis |ψ>? (since it escapes the bra-ket)
3. At 11:11, it seems like the LHS of the two equations stay the same but the RHS of the second equation got exponentiated?
and I might be mistaken here, but shouldn't an integration remove all of the "δ" in the equation, including |ψ+δψ>?
1) I see what you mean. The way to interpret the action operator inside that inner product is this: the operator acts like a differential operator (on the right state, but also on the left state). Let me remind that this is a "heuristic" derivation, so the notation is not always rigorous here. Any books on QFT that deal with the Schwinger formalism should cover this more rigorously.
2) "delta" represents the variation of the state due to a small change of the field. In general, the state changes if we perturb the field, so I would not consider your interpretation correct. "delta" can act both on the "bra" and on the "ket".
3) yes, at 11:11 we have a simple exponentiation.
The variable of integration on the left is the entire inner product between the "bra" and the "ket" you see in the expression. The result should be the logarithm as written in the video (the delta in the inner product does not disappear)
Another way to think about your first question. Consider the evolution of the change in the action between two states A and B. We can write the inner product: < B | δS | A >.
Based upon the considerations in the video, δS should depend only on the final and initial states, so it could be written as:
δS= S_B - S_A (where S_B is an operator that acts only on state B, and S_A acts only on state A).
Let's assume C is another state (that you can think of as being "in between" A and B). You can write:
δS = S_B - S_A = ( S_B - S_C ) + ( S_C-S_A )
Therefore:
< B | δS | A > = < B | ( S_B - S_C ) | A > +
< B | ( S_C - S_A ) | A >
Now, those differences in the action are going to create some small perturbations. In particular, S_B - S_C will act on < B |, whereas
( S_C - S_A ) will act on | A >.
This serves as another "justification" of the formula you were confused about.