Path Integrals in Quantum Theory

[rocdoc]

I have found a general result for certain exponential integrals that may be of interest to those involved with using path integrals. I am not certain that I am applying it correctly but it appears to work, and I can reproduce results quoted in various textbooks , using it. This may however be coincidental.

The result is ,result 7.4.32 of Abramowitz and Stegun, see pg.303 of Reference 1. I quote
'$$\int e^{-(ax^2+2bx+c)}\:\mathrm{d}x=\frac{1} {2}\sqrt{\frac{\pi} {a} }e^{ \frac{b^2-ac}{a}} erf(\sqrt ax+\frac{b} {\sqrt a})+const. ~~~~~~~~ (a\neq0) $$'here erf(z) denotes the error function.

Naively I take the following as true
$$\int^\infty_{-\infty} e^{-(ax^2+2bx+c)}\:\mathrm{d}x=\sqrt{\frac{\pi} {a} }e^{ \frac{b^2-ac}{a}}~~~~~~~(1)$$I assume this is true for complex valued $a,b~\text {and}~ c$

Please note Spiegel, Reference 2 pg.183 result 35.3 gives $erf(\infty)=1$ and Reference 1 pg. 297 says $erf(-z)=-erf~~z$

Using equation(1) , I can reproduce equation 8.18 of Kaku, see Reference 3; equation 1.1 of Bailin and Love, see Reference 4 and equation 1.49 of Cheng and Li , see reference 5.

So , am I OK with using equation (1)? Is it valid?

References

1) Handbook of Mathematical Functions , Eds M.Abramowitz and I.A. Stegun , Dover Publications, Inc., New York , Ninth Printing , Nov. 1970.

2) M. R. Spiegel, Ph.D. , Mathematical Handbook , McGraw-Hill , Inc. , 1968.

3) M.Kaku , QuantumField Theory, A Modern Introduction , Oxford University Press, Inc. , 1993.

4) D. Bailin and A.Love , Introduction to Gauge Field Theory, IOP Publishing Ltd, 1986.

5) Ta-Pei Cheng and Ling-Fong Li , Gauge theory of elementary particle physics ,Oxford University Press , New York, 1988.

 

[jambaugh]

It certainly should be valid with the understanding that you are extending it to an improper path integral in the complex plane. Since there are no poles there are no issues with choices of path and the result holds for limits at $\pm \infty + 0 i$.

 

[rocdoc]

Result 7.1.16 of Abramowitz and Stegun, Reference 1 of post1 , looks relevant to working out when the definite integral of post1 can be used. It is, I quote '

$$7.1.16~~~~~~~~~erf~ z \rightarrow 1~(~z\rightarrow \infty~~in~~~~|arg~z|<\frac{\pi} {4})$$

'.

 

[rocdoc]

To the result 7.1.16 of Abramowitz and Stegun, Reference 1 of post1 ,one may add
$$~~~~~~~~~~~~~~~erf~ z \rightarrow -1~(~z\rightarrow -\infty~~in~~~~|~\pi - arg~z|<\frac{\pi} {4})~~~~~~(2)$$
by use of symmetry, i.e.
$$erf(-z)=-erf~~z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3)$$
So I guess EQ(1) should be applicable, if
$$~~~~~~|~ arg(~\sqrt ax+\frac{b}{\sqrt a}~)|<\frac{\pi} {4}~~x\rightarrow \infty~~~~~(4)$$
and
$$~~~~~~|~\pi - arg(~\sqrt ax+\frac{b}{\sqrt a}~)|<\frac{\pi} {4}~~x\rightarrow -\infty~~~~~(5)$$
I also guess that the above inequalities may be extended to less than or equal versions, by the use of a suitable limit definition of $erf ~z$ for the boundaries of the regions of the complex plane mentioned in inequalities (4) and (5).

So with both $a$ and $b$ purely imaginary and 'positive' ( i.e. equivalent to points on the positive y-axis, in the complex plane), I guess we are OK to use the expression in EQ(1) for the definite integral, and also for $a$ and $b$ purely imaginary with $a$ positive but $b$ negative. I have not checked out the $a$ negative cases.

It also looks that it is OK to use the formula of EQ(1) for $a,b,c$ all real, which may account for result 15.75 of reference 2, which I quote'
$$15.75~~\int^\infty_{-\infty} e^{-(ax^2+bx+c)}\:\mathrm{d}x=\sqrt{\frac{\pi} {a} }e^{ (b^2-4ac/4a)}~~~~~$$
Obviously the above is not a complete analysis of where EQ(1) may be applicable. I would be interested in a more complete analysis of when EQ(1) can be used.