Final round of this thread!
We will evaluate the path integral (PI) using an approximation known as “saddle point”. It is a semiclassical approximation sometimes referred as the method of steepest descent. Moreover, it is based on the Lagrange method. Firstly, we take the integral
where is any function with (likely) several local minima (maxima). The constant can be take both real or complex, in particular purely imaginary!
Suppose now that we are interested in the limit of this integral when or equivalently (in the complex case) when . The, the integral will be dominated by the minima of S. In that case, we can aproximate it by a series of gaussian integrals, one for every minimum of S. If is such a minimum, then, in its neighbourhood, we obtain an expansion
and we can write
As a simple example, we consider again the harmonic oscillator perturbed by an anharmonic term quartic in q, i.e., we take an harmonic oscillator plus an interacting term with quartic power in the potential. The propagator reads, of any initial and final points (take the mass equal to one for simplicity):
Now, we define the PI for the harmonic oscillator with a source term (physically, it could describe the action of an external force) added to the lagrangian as follows:
This PI can be evaluated exactly. Now, if we choose such as
we get for the first order correction the propagator
In the last term above, we take four functional derivatives of and then we set , and thus only the piece of , quartic in J, is relevant. Fewer than four J’s will be killed (annihilated) by the derivatives, more than four will be killed when we set . That is,
where we have introduced the notation
Substituting the last compact expression into the actual PI, we obtain the formula
and now,working out a little bit more
then the first derivative can act either on or . It provides a delta function in any of these two cases and it reduces the integral to
In fact, the remaining derivative acts similarly an dills the remaining integral, and the result simplifies
Thus, the functional derivatives above, as a generalization of this simple result, provide
and again, we get
The time integral is just the time interval , and the final result of this long calculation can be written at last as follows
Please, note the dependence in in the propagator at first order in . It is a very different behaviour to the mere free particle or the simple harmonic oscillator!
QFT and field theory path integrals
It is very easy to generalize the PI to many, finite or even infinite, degrees of freedom. Field theory is indeed a “complex” or complicated system with infinite degrees of freedom. We turn particle positions into field functions:
and it constitutes a continuous (infinitely coordinated) system (). The analogue of the quantum mechanical propagator is the transition amplitude to go from one field configuration at to another at :
where is the field action, for instance, for some scalar field we could choose arbitrarily. Take, e.g., the field theory action
The field theory PI captures ALL the possible field configurations obeying the stated initial and final boundary conditions.
In field theory we are also interested in the Green functions. Quantum Field Theory calculations require the computation of the next object (the correlation functions!):
where the terms inside the vacuum expectation value are quantum operators, operator valued quantities of coordinates (and momenta). The vacuum expectation value (v.e.v.)= of any time-ordered product of Heisenberg field operators are very important quantities. In fact, this quantity IS known as the Green function or the correlation function. Motto to remember in QFT:
This lemma provides interesting connection and links, and it is a useful reminder or dictionary between different “worlds” that are, secretly, the same thing.
The order of the operators is such that the earliest field is written last (right-the most), the secon earliest second last, and so on. Example:
Green functions are related to amplitudes for the physical processes sush as scattering and decay processes in particle physics as well. But this is a long story some day I will tell to you!
Examples from field theory
Example 1. Free scalar field.
In the case of the free scalar field, the generating functional is given by
Both numerator and denominator can be expressed in terms of PI. The numerator reads
where is the classical configuration and are deviations or quantum fluctuations of the field configuration. Then, we integrate out over the deviations from the classical configurations. The action can be proved to be
where we neglected the linear terms in since they are reduced to zero due to the equations of motion. Thus, we get
Note that C is independent of J and will cancel in the generating functional Z(J).
Usint the fact that the classical field configuration satisfies the equations of motion (EOM)
Finally, we can write as well
i.e., the propagator of the Klein-Gordon field theory (Green function), since
The Green functions are found by solving its equation in 4-momentum space, and it yields
Let me calculate the 2-point Green function corresponding to this system:
If we expand the generating functional in powers of J, we obtain
The term quadratic in J is the only one that survives in the integral, and thus
There are two identical terms inside this last expression, depneding on which derivative acts on which J. The final result is simple
Therefore, the Green function (two point correlation function) int the QFT is also the Green function in the usual differential equation sense! It is a cool (consistent!) result! Lut us compute the 4-point correlation function further
The only term that survives is the part of the exponential that contributes with four J’s. Thus, the 4-point correlation function for the free particle reads
There, there are terms, corresponding to the number of ways of associating the derivatives with the J’s (sources, quantum operators). In 8 of them, the Green functions which arise are , and so on. The final result reads (you can check it with the use of the Wick theorem!):
And the cool thing is that this thing can be represented diagrammatically in a convenient way related to Feynman graphs! That is the hidden power of QFT!
Example 2. Interacting field theory.
Unfortunately, it is not possible to compute exactly the generating functional Z(J) in general. It is necessary to use approximate methods or sometimes, systems with a big symmetry to compute any exact Z(J). Let us use perturbation theory, and we consider a scalar field theory with interactions. This is called the theory as well. It is defined by a lagrangian density
Then, the generating functional reads
In the next step, we change the higher-order term by a functional derivative with respect to J:
We can push out the first exponential from the integral and the remaining functional derivative is that of . We adjust the constant C so that and we get
We can write a perturbative expansion for any Green function/propagator/correlation function from this last expression!
The perturbative expansion for the two point Green function is “simple”. Let me look at to the first non-zero order in . We find that
where the brackets imply integration over the positions of the J’s. In both numerator and denominator, we can expand both exponentials and the only remaining terms are those which the same total number of derivatives and J’s. Let me look at the linear term in . There are 6 functional derivatives, so we need the term from the expansion of the second exponential with 6 J’s. For this term, we get the following expression in the numerator
There are now 720=6! terms. However, there are only two distinct analytical expressions in the result. The first of these arises if the functional derivatives at and act on different brackets. Combinatorics tells us that there are 576 of such terms, yielding the expression
The only different term to this one arises when the derivatives at and act on the same bracket. This accounts for terms (note that 720=576+144). The analytic form of this term reads
The denominator can be evaluated in a similar way. The Green function to order is finally recasted into
Since we have only computed the numerator and denominator up to order , we can rewrite this last expression in the following manner:
We can now cancel the second factor in the numerator against the denominator, as you can easily see, up to order , and we get as final ultimate result that
This factorization of the numerator into a part containing no factors independent of the external position times the denominator occurs to all orders, as you can be prove yourself easily if you have understood these notes. Thus, we can define the “disconnected” parts of the Green function as the parts of the diagrams not connected to any external line. The conclusion is that this disconnected parts cancel from the Green functions!
Additional exercises for you
Exercise 1. Show that if a functional is given by any expansion of the type
then you can get the functional derivative
Exercise 2. Compute, using the definition of functional derivative you have studied in this thread, the functional derivatives of
b) , with a fixed function, and IF
is the functional derivative.
Exercise 3. Compute explicitly the integral
where is any complex, symmetric matrix with eigenvalues such as .
Exercise 4. Show that, for a gaussian distribution function, the generating function IS
Exercise 5. Compute the n-point functions for the harmonic oscillator (i.e., the 1-point, 2-point,…correlation functions).
Exercise 6. Compute with the saddle point approximation the integral corresponding to
in the case that a) , i.e., imaginary coupling constant, and b), where the coupling constant is real or complex in general. Compare the results with the results of a perturbation expansion assuming that , i.e., with the result of a tiny anharmonic term.
Exercise 7. In quantum field theory with interacting coupling, compute the Green function (2-point correlation function) up to order . Compute the 4-point correlation function (Green function) to order . Compare the expressions you obtain with the results given in this thread.
Exercise 8. In QFT with a interaction term of type , compute the 2-point Green function to order and the 4-point correlation function to the same order. Compare the results with those of the theory.
Some references for further study
Let me finish this interesting thread with some basic and well known references:
1) The Dirac original idea for the role of the action in QM is inside the article
Physikalishche Zeitschrift der Sowjetunion, Band 3, Heft 1 (1933). P.A.M.Dirac.
2) Feynman pioneer works about the path integral are mainly here
i) Reviews of Modern Physics 20, 367.(1948). R.P.Feynman.
ii) Quantum Mechanics and Path Integrals, McGraw Hill, 1965. R.P.Feynman and A.R.Gibbs.
iii) Statistical Mechanics: A set of lectures, 1972. R.P.Feynman. Benjamin eds.
3) Some classical works about the path integral:
i) Lectures in Quantum Mechanics, Benjaming-Cummings, 1973. G.Baynn.
ii) L.S.Schulman, Techniques and Applications of Path Integration, John Wiley and Sons, 1981.
4) Field theory and path integrals are treated in some basic texts about this hard subject. For instance:
i) Field theory, renormalization group and critical phenomena, 2nd edition, World Scientific eds. 1984. D.J.Amit.
ii) L.H.Ryder. Quantum Field Theory, Cambridge University Press, 1985.
iii) P.Ramond, Field Theory: a modern Primer, 2nd edition. Addison-Wesley, 1990.
iv) E.S.Abers and B.W.Lee, Physics Reports 9C, 1,1973.
v) B.Sakita, Quantum Theory of Many Variable Systems and Fields, World Scientific, 1985.
5) More advanced topics in QFT can be found here:
i) S.Coleman, Aspects of Symmetry, C.U.P. 1985. I own and admire this book, written by one of the fathers of the wormhole concept and its application into physics!
ii) Quantum Field Theory in Condensed Matter Physics, Cambridge University Press (C.U.P.), 1995 by A.M.Tsvelik (purely soviet and russian style!).
iii) V.N.Popov, Functional integrals and Collective Excitations, C.U.P. 1987. Other great book by one of the best russian QFT theorists.
iv) Field Theories of Condensed Matter Systems, Addison-Wesley, 1991. E.Fradkin. A great book describing the links between Statistical Mechanics, Condensed Matter and the methods of field theory, by one of the best masters in the subject!
Today, you can find even more modern and better lectures (in some cases, not always) in the internet. But the above references are classical and deserve at least a minimal fast reading.
See you in my next SPECIAL AND SUPER(HYPER)GEEKNERD TSOR post, the 150th!!!!!