This note gives a comparison on the three quantum pictures. Also, the Dyson series are introduced with the help of time-ordering operators.

1. Comparison of three pictures

#### 2. Evolution Operator

The evolution operator in Shrodinger picture is defined as

(1)

In Interaction picture, we can also defined a evolution operator

(2)

#### 3.Dyson Series

Combine Schrodinger equation and the definition of evolution operator, we can get the equation for evolution operator

(3)

As for the solution of the above equation, there are 3 cases.

Case 1. is independent of time

(4)

Case 2. is time-dependent but the ‘s at different times commute

(5)

Case 3. ‘s at different times do not commute

(6)

which is just the Dyson series.

As for the interaction picture, we have similar results.

(7)

Case 1. is independent of time

(8)

Case 2. is time-dependent but the ‘s at different times commute

(9)

Case 3. ‘s at different times do not commute

(10)

#### 4. Time Order

Introducing the time-oder operator, the Dyson series can be written in a more compact form. The time-order operator is defined as

(11)

where and is the number of permutations.

In our problem, the s in Eq.6 or s in Eq.10 are all time-ordered. It’s always even number of Fermionic fields appear in the and . There is no an extra minus sign for the time ordering.

(12)

Also, if we define

(13)

and is symmetric in its arguments, we can divide the integration region into sub regions, in which , , etc. They all equals to

(14)

thus,

(15)

Then, Eq.10 can be written as

(16)

can be written out in a similar way.

#### 5. Correction on Section 4

Section 4 is wrong. Exactly, we don’t need to use the symmetry properties of the integrand. On the contrary, the time-ordering operator is introduced in case that the integrand is not symmetric on s.

Take the second order term in Eq.6 for example. The integration area is the blue shadow area in the following figure. We can change the order of integration with the value of integration unchanged since it’s still the same integration area. Thus,

(17)

Notice that above. We now change dummy variables in the second term, interchanging the labels and , and the second term becomes

(18)

notice that on the RHS and the integration area becomes the red shadow area, but we still have the later time on the left. Thus

(19)

It’s an integration on the hole area with the integrand is time-ordered.

Similarly, it can be generalized to higher order.