Functional integrals & quantum field theory

solution which arise as corrections in ¹{^¯h. The solution to the saddle point equation are the classical equations of motion (2.18) that minimizes the classical action. Thus, we can expand about the saddle point solution up to quadratic order in the fluctuations

Srqs «Srqcls `1 2 ż

dtdt¹pq´qclqptq δ²Srqs δqptqδqpt¹q

ˇˇ ˇˇ

q“qcl

pq´qclqpt¹q. (2.19) Note that Srqsis strictly a functional and so we need to apply second order functional deriva-tives. The first order functional derivative ofSrqsvanishes when evaluated on solutions of the equations of motion q_cl. Note also that the second order functional derivative is positive defi-nite since the classical solution minimizes the action and hence the convergence of the Gaussian integral is ensured. The stationary path integral reads then

ż

qf“qptq qi“qp0q

Drqse^i{¯hSrqs “ÿ

ℓ

e^{i{¯hSrqpℓqcl s} det

˜ i 2π¯h

δ²Srq^pℓqs δq^pℓq_f δq_i^pℓq

¸´¹{²

, (2.20)

where the sum over ℓ runs over the contribution from different saddle point solutions. For

¯

h Ñ0 this is the dominant/leading term in the asymptotic expansion. To make the argument mathematically more rigorous one should start from the analytic continuation and expand to higher orders in the fluctuations. Then, the method of steepest descend ensures the convergence of the Gaussian integrals and enables us to choose the constant phase such that the derivatives of the function are real. For an even complex functionfpzqwe find

I“ ż

C

dz e^tfpzq « d 2π

tf²pzqexp

„

tfpz0q `3f^pivqpz0q tf²pz0q



, (2.21)

where we see that higher order terms are suppressed with higher powers of t. This power counting carries over to non-Gaussian integrals arising from higher order fluctuation terms in the actionSrqs.⁴ In general we would find contributions to the path integral (2.20) scaling with

ż

qf“qptq qi“qp0q

Drqse^i{¯hSrqs „exp

«

¯ h^´1

˜

Srq_cls ` ÿ

ną2 neven

¯

h^n{2 δⁿSrqs δqⁿ

Nδ^n/2Srqs δq^n/2

ˇˇ ˇˇ

q“qcl

¸ff

, (2.22)

wherendenotes the (even!) expansion order in the fluctuation. Again for¯h Ñ0the classical action is the leading term in the asymptotic expansion. Since this is an asymptotic expansion, we can only truncate the series for systems where quantum fluctuations are small. With this final remark we conclude our discussion on path integrals. For a more in-depth discussion and detailed insight see the excellent textbooks [44,45].

Classical Mechanics

Quantum Mechanics

Classical Field Theory

Quantum Field Theory averaging procedure

8degrees of freedom

pathintegral formalism

coherent states functional integrals

canonical quantization quantum statistics

imaginarytime

Figure 2.1. Various connection between the corner stones of theoretical physics. Formal sim-ilarities and analogs in mathematical representation allows for interconnections and methodical transfer between conceptual different physical frameworks. One of the most fruitful and interesting connection is the mapping of quantum statistics onto (classical) field theory.

footing.⁵ This gives way to a formulation where the quantum mechanical structure is imple-mented in field valued operatorsϕpt,xqdescribing the creation/annihilation of excitations from a vacuum state at given timetand pointx, in close analogy to wavelike excitations in classical field theory (see Figure2.1) The right eigenstates of the annihilation operators

a

k are called coherent states

|ϕy “exp

˜ÿ

k

ϕk

a

_k^:

¸

|0y,

a

k|ϕy “ϕk|ϕy @k. (2.23) Note that the “excitation number” is not constant and thus we cannot have an eigenstate for the creation operator. It is possible to take the Hermitian conjugate of (2.23) to define a left eigen-state for the creation operator

a

^:. So for the functional integrals we will write the Hamiltonian in the coherent state representation replacing the momentum and position eigenstates of the path integral. Following the same steps

ii

_to

iv

_{on page15}as for the derivation of the path integral we find the functional integral for quantum fields ind`1dimensions. The conceptional difference lies in the interpretation of the transition amplitude. Now all calculations are done with respect to the vacuum state|0y which removes our knowledge of the preparation of the state and thus the boundary conditions of the initial and final state are removed

xϕfpxq;tf|ϕipxq;tiy “ ż

ϕpti,xq“ϕipxq ϕptf,xq“ϕfpxq

Dϕe^iSrϕs ÝÑ x0;8 |0;´8 y “ ż

Dϕe^iSrϕs. (2.24) Note that we now need to use the Heisenberg picture where the state vectors are time indepen-dent in order to compare to time depenindepen-dent field operators, so we end up with the vacuum to vacuum transition amplitude attÑ ˘8. We see that the propagator is now a correlation func-tion of two field operators inserted at the spacetime points where excitafunc-tions are created and

5This choice amounts to a quantum mechanical unitary position operator. One might wonder if there could be a way to promote the classical parameter timetto a true Hermitian quantum operatortˆ(implying that time becomes measurable). This leads us in the realm of “quantum spacetime”. Interesting ideas (also about the different meanings of time) can be found in [46].

annihilated

xqf;tf|qi;tiy ÝÑ x0;8 |ϕptf, xfqϕpti, xiq |0;´8 y. (2.25) Furthermore, we can view quantum mechanics as a 0`1 dimensional quantum field theory, where the pair of conjugated creation and annihilation operatorst

a

,

a

^:uis mapped to the conju-gated position and momentum operatort

q

,

p

uby a canonical transformation.

Correlation functions and generating functionals

In quantum field theory we are concerned with the correlation functions (or correlators) of operators inserted at different spacetime points. The information about the quantum system is encoded in the correlation functions of all physical operators. The correlation functions are defined as the functional expectation value of a product of operatorsϕpxq⁶

xϕpx₁q ¨ ¨ ¨ϕpx_nq y “ 1 Z

ż

Dϕ ϕpx₁q ¨ ¨ ¨ϕpx_nqe^iSrϕs. (2.26) Note that by construction the functional integral is time ordered so a time ordering operator is implicitly assumed in all expectation values. The functionalZis called partition function, which encodes the vacuum structure of the theory, an is used as a normalization factor. As we will see in Section2.1.4it is the primary object of statistical field theory. Here it ensures the right normalization of the expectation value

Z“ ż

Dϕe^iSrϕs ô x1y “ 1 Z

ż

Dϕ1e^iSrϕs “1. (2.27) All correlation functions can be constructed by taking functional derivatives of the so-called generating functional which is the partition function with an additional source term preparing the system in the state ϕpxq. The preparation is usually done by an external device (a filter projecting the quantum state onto the particular state the system is prepared in) described by the external source fieldJ. Therefore, the full actionSrϕ;Jsis described by an additional interaction between the fieldϕand its sourceJ

ZrJs “ ż

Dϕexp

„ i

ż

d^dxpL `ϕJq



“ ż

Dϕe^iSrϕ;Js. (2.28)

In principle there are various ways to prepare the system so there are different source fieldsJ˜that define different generating functionalZrJ˜s. But any measurements described by an observable must be independent of the specific preparation (e.g. a specific experimental setup) and thus we need to setJ“J˜“0to obtain an objective observable

xϕpx1q ¨ ¨ ¨ϕpxnq y “ p´iqⁿ Zr0s

δⁿZrJs δJpx1q ¨ ¨ ¨δJpxnq

ˇˇ ˇˇ

J“0

, (2.29)

where the functional derivative is defined as δJpxq

δJpyq “δpx´yq ÝÑ δ δJpyq

ż

d^dx Jpxqϕpxq “ϕpyq. (2.30)

6In the following we will use units such that¯h“1.

The correlation functions (2.29) encode, as their name suggest, the correlations between two field operators at different spacetime points. If the amplitudes of the fields fluctuate indepen-dently then the correlation function will vanish. Due to the non-local nature of quantum fluc-tuations, non-zero correlation functions describe the entanglement of the respective operators.

There are two special correlation functions, namely the vacuum expectation value of a field xϕpxq yand the propagatorxϕpxqϕpyq ydescribing the propagation of the influence of the oper-ator over the range|x´y|. A prominent example is the creation of a particle at the spacetime pointxand the annihilation at the spacetime pointy which amounts to a particle propagating formxtoy. The correlation functions defined so far include also lower order correlations which are (topologically) disconnected from each other describing lower order entanglement. The log-arithmic derivative of the partition function removes these disconnected parts such that only connected terms remain. We therefore define the connected correlation function as

xxϕpx₁q ¨ ¨ ¨ϕpx_nq yy “ p´iqⁿ δlnZrJs δJpx1q ¨ ¨ ¨δJpxnq

ˇˇ ˇˇ

J“0

. (2.31)

For instance the propagator can be split into

xxϕpxqϕpyq yy “ xϕpxqϕpyq y ´ xϕpxq y xϕpyq y

“@ `

ϕpxq ´ xϕpxq y˘`

ϕpyq ´ xϕpyq y˘ D

, (2.32)

where we explicitly see that we removed the lower order vacuum expectation value. Therefore, the connected propagator really describes the field excitations above the vacuum. This can be generalized to include the so-called quantum chain rule, see Appendix B.2. In this case the quantum chain rule is not needed because our fields in the path integral are mere complex numbers so we can commute them in an arbitrary fashion. This is also the reason why the functional integral yields a time ordered correlation function.

Gaussian integrals and Wick’s theorem

Now let us look at a very simple case that can be solved explicitly using Gaussian integration.

In fact a lot of techniques used in field theory rely on the transformation of a more complicated functional integral into a Gaussian integral which can be solved exactly. Completing the square, we can solve integrals of the type

ż

Dϕexp

„ i

ˆ1 2 ż

d^dxd^dy ϕpxqG^´1px, yqϕpyq ´ ż

d^dx ϕpxqJpxq

˙

“Na

detpiGqexp ˆ1

2 ż

d^dxd^dy JpxqGpx, yqJpyq

˙

. (2.33) In the special case J “ 0 the integral reduces to the square root determinant of the operator kernel (or Green function)G. The determinant is understood as a determinant of an Hermitian operator.⁷ Since we are dealing with a bilinear form we can take functional derivatives of (2.33) with respect toGpx¹, y¹q^´1andJ “0. The left hand side gives rise to

i 2

ż

Dϕ ϕpx¹qϕpy¹qexp ˆi

2 ż

d^dxd^dy ϕpxqG^´1px, yqϕpyq

˙

“Na

detpiGqi 2

@ϕpx¹qϕpy¹qD (2.34)

7In principle one needs to be very careful to distinguish between self-adjoint and Hermitian operators and to make sure that there are no residual contributions (c.f. also the discussion in AppendixB.3).

and the right hand side (see AppendixB.1) N δa

detpiGq δGpx¹, y¹q^´1 “N i

2

adetpiGqGpx¹, y¹q

Equating (2.34) and (2.35) yields

@ϕpx¹qϕpy¹qD

“Gpx¹, y¹q (2.35)

Acting again with the functional derivative^δ{^δG´1on (2.35) we obtain

xϕpx1qϕpx2qϕpx3qϕpx4q y “Gpx1, x2qGpx3, x4q `Gpx1, x3qGpx2, x4q

`Gpx1, x4qGpx2, x3q. (2.36) For more technical details (using discretized matrices) see AppendixB.1. Taking the derivatives with respect to G´1 can be iterated to generalize (2.36) to express arbitrary even correlation functions in terms of all possible pairings of “contracted” two-point correlation function

xϕpx1q ¨ ¨ ¨ϕpx2nq y “ ÿ

Ptpairings ofpx1,...,x2nqu

Gpxi1, xi2q ¨ ¨ ¨Gpxi_2n´1, xi2nq. (2.37)

Note that all correlation functions with odd insertions vanish due to the structure of a bilinear form. As a matter of fact they cannot be generated by virtue of the symmetric/Hermitian form of the operator. Using the full form (2.33) and taking functional derivatives with respect toJpx¹q (as in the prescription (2.29)) we see that all odd correlation functions must vanish since an odd integrand is integrated to zero over a symmetric rangep´8,8q.

This only holds true for the free theory. As explained in the next Section2.1.3 we can include interactions by looking at perturbations. All these concepts can be derived for fermions as well.

Due to the anti-commuting nature of fermions one needs to introduce Grassmann numbers and Grassmann valued fields. Since we will use field theoretical methods in the context of effective field theories for critical phenomena, we will not have to deal with fermionic fields since order parameters or condensates are bosonic objects due to the strange property that fermions are always excited/created in pairs. So the total number of fermions must be an even integer. This non-local constraint gives rise to non-local excitations [4]. This “pairing” constraint also prohibits any macroscopic object such as a condensate to behave like a truly fermionic object. Therefore, macroscopic observables must be bosonic.

Im Dokument Gauge/Gravity duality (Seite 39-43)