Understanding Convergence Factors in Physics Textbooks

[maverick_starstrider]

"Convergence Factors"

In all my textbooks I always see these random convergence factors thrown in (+0's or +i*nu or some such) but I have never seen a book that would dirty itself by steeping so low as to explain what they are (I'm looking at you Wen, Bruus and Flensberg, Fetter and Walecka, etc.). I understand they are supposed to be some ad hoc infinitesimal added to guarantee some integral converges but can anyone point me to (or provide) an explanation of:

-when they are necessary
-why it is ok to add them on a whim
-what would happen is we didn't add them
-what is the physical meaning of adding them.


If you don't know what I'm talking about I'll give the example of the energy propogator

$G_E(n_b,t_b,n_a,t_a) = -i \langle n_b \vert U(t,t_0) \vert n_a \rangle = -ie^{-i \epsilon_n (t_b - t_a) } \delta_{n_b,n_a}$

where when we move to frequency space we get

$G_E(n_b,n_a,\omega) = \int_0^{\infty} dt G_E(n_b,t_a+t,n_a,t_a)e^{it\omega - 0^+ t} = \frac{1}{\omega - \epsilon_{n_a}+i 0^+} \delta_{n_b,n_a}$

I don't get the $0^+$.

Thanks in advance

 

[A. Neumaier]

In all my textbooks I always see these random convergence factors thrown in (+0's or +i*nu or some such) but I have never seen a book that would dirty itself by steeping so low as to explain what they are (I'm looking at you Wen, Bruus and Flensberg, Fetter and Walecka, etc.). I understand they are supposed to be some ad hoc infinitesimal added to guarantee some integral converges but can anyone point me to (or provide) an explanation of:

-when they are necessary
-why it is ok to add them on a whim
-what would happen is we didn't add them
-what is the physical meaning of adding them.


If you don't know what I'm talking about I'll give the example of the energy propogator

$G_E(n_b,t_b,n_a,t_a) = -i \langle n_b \vert U(t,t_0) \vert n_a \rangle = -ie^{-i \epsilon_n (t_b - t_a) } \delta_{n_b,n_a}$

where when we move to frequency space we get

$G_E(n_b,n_a,\omega) = \int_0^{\infty} dt G_E(n_b,t_a+t,n_a,t_a)e^{it\omega - 0^+ t} = \frac{1}{\omega - \epsilon_{n_a}+i 0^+} \delta_{n_b,n_a}$

I don't get the $0^+$.

Thanks in advance

The Fourier transform of distributions is not completely straightforward since the integrals don't make sense without the factors.

The +i eps (or +i 0) tells the way the integration contour must be deformed into the complex plane in order to get a well-defined distribution. See, e.g., http://en.wikipedia.org/wiki/Propagator_(Quantum_Theory)#Relativistic_propagators

They are not added in an ad hoc way but in order to get a free field theory with a valid unitary representation of the Poincare group satisfying causal (anti)commutation rules
(or, in condensed matter theory, the correct nonrelativistic propagator).

 

[maverick_starstrider]

I appreciate the response but I'm afraid I'm not getting much from that. Why aren't the Fourier transforms of distributions straightforward (and which factors). What is a "well-defined" distribution? What is our goal here with this integral, why do these integrals needs special contour deformations that others don't?

 

[A. Neumaier]

I appreciate the response but I'm afraid I'm not getting much from that. Why aren't the Fourier transforms of distributions straightforward (and which factors). What is a "well-defined" distribution? What is our goal here with this integral, why do these integrals needs special contour deformations that others don't?

What is your math background? It doesn't make sense to answer your questions before you get the relevant background knowledge.

Do you know what a distribution is?

Can you make sense of the integral without the +i eps? At some point the integrand becomes infinity - do you know how to define integrals in this case?

 

[maverick_starstrider]

What is your math background? It doesn't make sense to answer your questions before you get the relevant background knowledge.

Do you know what a distribution is?

Can you make sense of the integral without the +i eps? At some point the integrand becomes infinity - do you know how to define integrals in this case?

I understand the math in the sense that you're essentially saying the integral is valued at its Cauchy Principal Value (am I wrong?). It's the physics and motivation of this. WHY is it physically correct to integrate around our pesky poles (or push them off the real line). Why should this be considered the "true" value of the integral. It's really not the same integral at all is it? Whenever physics throws a function (or distribution) which must be integrated across one of its poles why is it ok to just push the pole out of the way? This is what I don't understand.

 

[A. Neumaier]

I understand the math in the sense that you're essentially saying the integral is valued at its Cauchy Principal Value (am I wrong?). It's the physics and motivation of this. WHY is it physically correct to integrate around our pesky poles (or push them off the real line). Why should this be considered the "true" value of the integral. It's really not the same integral at all is it? Whenever physics throws a function (or distribution) which must be integrated across one of its poles why is it ok to just push the pole out of the way? This is what I don't understand.

The usual intuitive justification is that the small systems considered in scattering experiments are not truly isolated but coupled to the environment, so that a tiny bit of energy dissipates. This makes the Hamiltonian slightly nonhermitian (adds an optical potential). But of course it os too small to be worth modeling it directly. so one treats the system as conservative and only adds the infinitesimal i eps.

A more rigorous justification is obtained if you start with axiomatic properties that must be reasonably assumed to hold for any decent field theory, based on relativity, causality, and stability. This leads to the Wightman axioms. Then one can show (see Weinberg) that the free fields satisfy the axioms only if the poles are treated in the textbook fashion.

On a more elementary level, one can see how the i eps arises rigorously for nonrelativistic problems in the solution of the Lippmann-Schwinger equation. For a fairly rigorous discussion, see Vol. 3 of Thirring's Course on Mathematical Physics.