For simplicity we will start with a one-dimensional Hamiltonian H= ˆp2
2m +V( ˆx) , ˆp= ¯h i
∂
∂x, (6.1)
where we have indicated a hat on top of operators to distinguish them from number-valued coordinate and momentum. We wish to study the time-evolution operator exp(−i HT/¯h). In the coordinate representation its matrix is given by
<x�|e−i HT/¯h|x >, (6.2) where|x >is the position eigenfunction. We will also need the momentum eigenfunction|p>, i.e., ˆp|p>= p|p>, whose wave function in the coordinate space is given by
<x|p>= ei px/¯h
√2π¯h. (6.3)
Indeed, one verifies that
¯h i
∂
∂x <x|p>=p<x|p> . (6.4) An important ingredient in deriving the path integral expression will be the completeness relations
ˆ1=
d x|x><x| and ˆ1=
dp|p>< p|. (6.5) For arbitrary N we can use this to write
<x�|e−i HT/¯h|x>=<x�|
e−i HT/N¯hN
|x>
=
· · ·
<x�|e−i HT/N¯h|xN−1>d xN−1
<xN−1|e−i HT/N¯h|xN−2>d xN−2
<xN−2| · · · ·e−i HT/N¯h|x2>d x2
<x2|e−i HT/N¯h|x1>d x1<x1|e−i HT/N¯h|x> . (6.6) 29
30 A Course in Field Theory We will now use the so-called Trotter formula
e−i( A+B)/N=e−i A/Ne−i B/N
1+O( N−2)
(6.7) for two operators A and B. This can be seen by expanding the exponents, and the error term is actually of the form [A, B]/N2. (One can also use the Campbell–Baker–Hausdorff formula, which will be introduced later). With the Hamiltonian of Equation (6.1) this can be used to write for N→ ∞
e−i HT/N¯h =e−i ˆp2T/2mN¯he−i V( ˆx)T/N¯h. (6.8) By inserting the completeness relation for the momentum we can eliminate the operators
This can be done for each matrix element occurring in Equation (6.6). Writing t=T/N, xN=xand x0=x we find
It is important to observe that there is one more p integration than the number of x integrations.
The integrals in the path integral are strongly oscillating and can only be defined by analytic continuation. As parameter for this analytic continuation, one chooses the time t. Fort = T/N ≡ −iT/N = −iτ, the Gaussian
Path Integrals in Quantum Mechanics 31 integral over the momenta is easily evaluated
∞ or after substitutingT =i T we find
<x|e−i HT/¯h|x >= lim This is the definition of the path integral, but formally it will often be written as
since the discretised version of the action with xj ≡x(t= jt) is precisely Sdiscrete=tN−1
It is important to note that the continuous expression is just a notation for the discrete version of the path integral, but formal manipulations will be much easier to perform in this continuous formulation. Furthermore, the integral is only defined through the analytic continuation in time.
However, if we integrate over xN = x0 this analytically continued path integral, with T = −iT, has an important physical interpretation
d x <x|e−HT/¯h|x>=Tr(e−βH)β=T/¯h. (6.16)
32 A Course in Field Theory It is the quantum thermal partition function (the Boltzmann distribution) with a temperature of ¯h/kT. In the continuous formulation we therefore have
Tr(e−TH/¯h)=
x(T)=x(0)Dx(τ) exp[−SE/¯h] (6.17)
in which SE is the so-called Euclidean action SE =
It is only in this Euclidean case that one can define the path integral in a mathematically rigorous fashion on the class of piecewise continuous func-tions in terms of the so-called Wiener measure
meaning that this measure is independent of the way the path is discretised, Figure 6.1.
For more details on this, see Quantum Physics: A Functional Integral Point of View, by J. Glimm and A. Jaffe (2nd ed., Springer, New York, 1987).
We will now do an exact computation to give us some confidence in the formalism. To be specific, what we will compute is the quantum partition function for the harmonic oscillator, where V(x)= 12mω2x2
ZN≡
Note that we have now N integrations, because we also integrate over x0 = x(0)=x(T)=xNto implement the trace. The path involved is thus periodic in time, a general feature of the expression for the quantum partition function
∆t t x
FIGURE 6.1 Contributing path.
Path Integrals in Quantum Mechanics 33 in terms of a path integral. We now rescale
yi=xi
to obtain the simple result ZN= We can diagonalise the quadratic term by using Fourier transformation
yk= 1
√N
N−1
=0
be2πik/N, b∗=bN−, b∗0=b0. (6.23) It is easy to verify that the Jacobian for the change of variables yi → b is 1, and one obtains a result that must look familiar from the classical small oscillations problem for a finite number of weights connected by strings,
ZN=
We can convert the product to a sum using a Laplace transform. We start with the identity We read off, from the definition of ZN, that Q is a sum of exponentials
Q(s, ˜ω)≡N−1
34 A Course in Field Theory This is a periodic function with period 1 [ fs(x+1) = fs(x)], and its discrete Fourier coefficients can be computed exactly
˜fs(k)= Please note that we have exchanged the sum overwith extending the inte-gration of x to the interval [0, N]. The last identity is one of the definitions of the modified Bessel function, see, e.g., Handbook of Mathematical Functions, by M. Abramowitz and I. Stegun (Dover, New York, 1978). The advantage of these manipulations is that the Laplace transform of this Bessel function is now (see the same reference)
∞
0
ds e−λsIν(s)=[λ+√
λ2−1]−|ν|
√λ2−1 , (6.31)
and as we can express fs(x) as a sum over these Bessel functions fs(x)=
k∈ZZ
e−2πikxINk(2s), (6.32) this allows us to evaluate Equation (6.27). For technical reasons, it is easier to compute the variation of the free energy with the frequency, where the free energy F is defined as
ZN( ˜ω)=exp[−βF ( ˜ω)]. (6.33) The geometric series is of course easily summed, but to make the result more transparent we introduce the scaled effective frequency
ωT/N=ωτ =ω˜ ≡2 sinh(12) (6.35)
Path Integrals in Quantum Mechanics 35 The last identity can be seen as the free energy of the harmonic oscillator with the frequencyN/T =/τ, as it can also be written as
τ
Amazingly, even at finite N the Euclidean path integral agrees with the quan-tum partition function of a harmonic oscillator, but with a frequency that is modified by the discretisation; see Eqs. (6.21) and (6.35). It is trivial to check now that the limit N→ ∞is well defined and gives the required result, since
N→∞lim N/T =ω. (6.38)
In general the exact finite N path integral is no longer of a simple form.
Nevertheless, one can evaluate this exact expression in relatively simple terms (which will verify the above result along a different route; see also Problem 10). So from now on, we will take the potential arbitrary and in a sense we follow the derivation of the path integral in the reverse order.
ZN(x, x;T)= This means that we can define an effective Hamiltonian by
e−H( N)T/¯h≡
But this Hamiltonian is not Hermitian as one easily checks from the above expression, since under conjugation the order of the exponents containing the kinetic and potential terms is reversed. This can be corrected in two ways
e−H1( N)τ/¯h ≡exp
36 A Course in Field Theory leading to two equivalent expressions for the finite N path integral
ZN(x, x;T)=<x|exp In particular the partition function is given by
ZN =
d x ZN(x, x,T)=Tr(e−H1( N)T/¯h)=Tr(e−H2( N)T/¯h). (6.43) It is actually not too difficult to show that there exists a unitary transforma-tion U, such that U H1U† = H2, which shows that both choices are indeed physically equivalent.
In principle we can now compute Hi( N) for finite N as an expansion in 1/N, by using the so-called Campbell–Baker–Hausdorff formula
eAeB =eF ( A, B), F ( A, B)=A+B+12 which is a series in multiple commutators of the, in general, noncommuting operators A and B. It can be derived by expanding the exponentials, but in the mathematics literature more elegant constructions are known, based on properties of Lie groups and Lie algebras. These objects will be discussed in Chapter 18. For the harmonic oscillator, working out the products of the exponential can be done to all orders and one finds (see Problem 10 for details)
τHi( N)= ˆp2
2Mi +12Mi2ˆx2, (6.45) withdefined as in Equation (6.35) and the effective masses Mi defined by
M1=2m tanh(12)
τ , M2= m sinh()
τ . (6.46)
One can now explicitly verify that
compare this to Equation (6.37) Tr Now we have seen that, at least for some examples, the limit of increasingly finer discretisation is in principle well defined, we can think of generalisation
Path Integrals in Quantum Mechanics 37 to an arbitrary number of dimensions (n) (for field theory even to an infinite number of dimensions).
We have purposely also given the expression that involves the path integral as an integral over phase space, as it shows that the Gaussian integration over the momenta effectuates the Legendre transform
ip· ˙x−i p2
2m −i V(x)= −i(p−m˙x)2
2m +im˙x2
2 −i V(x), (6.49) which is equivalent to the stationary phase approximation for the momentum integration
An other interesting example of the path integral is the case of the interac-tion of a charged particle with a magnetic field. In that case one has for the Hamiltonian
H( ˆp, ˆx) =( ˆp−eA( ˆx))2
2m +V( ˆx). (6.51)
Now, however, the matrix element<pi|exp(−i Ht/¯h)|xi>will depend on the specific ordering for the position and momentum operators in H. Different orderings differ by terms linear in ¯h, or
A( ˆx) ·pˆ =pˆ· A( ˆx)+i¯h∂iAi( ˆx). (6.52)
38 A Course in Field Theory So, by choosing the so-called Coulomb gauge∂iAi(x) = 0, the problem of operator ordering disappears. We leave it as an exercise to verify that the action, obtained from the Legendre transform, is given by
S=
}. Using the path integral this means that proves that the path integral derived from Equation (6.53) has the correct properties under gauge transformations, despite the fact that the derivation was performed by first going to the Coulomb gauge.
As long as the Hamiltonian is quadratic in the momenta, the stationary phase approximation for the momentum integral is exact. However, also for the coordinate integrals we can use the stationary phase approximation (exact for a harmonic oscillator), which is related to the WKB approximation in quantum mechanics. It gives a way of defining an expansion in ¯h, where in accordance to the correspondence principle, the lowest-order term reproduces the classical time evolution. Indeed the stationary phase condition
−δS
x. We expand around these solutions by writing There is no term linear in qi(t), as this term is proportional to the equations of motion, or equivalently to the stationary phase condition. For the simple Lagrangian L=12m˙x2−V(x) one has
Path Integrals in Quantum Mechanics 39 For the harmonic potential, V=12mω2x2, where the stationary phase approxi-mation is exact, i.e., there are noO(q3) corrections. Introducingq (t), however, splits the action in a classical piece that depends on the boundary conditions and a quantum piece described by a harmonic oscillator action for the fluctua-tions around the classical path that is independent of the boundary condifluctua-tions and the classical path
S(x) =S(xcl)+ T
0
dt (12m˙q2−12mω2q2). (6.59) In practical situations one splits from the action the quadratic part in the coordinates and velocities and considers the rest as a perturbation. In that case xclis the classical solution of the quadratic part only. As this can always be solved exactly, and as nonquadratic path integrals can rarely be computed explicitly, this will be the way in which we will derive the Feynman rules for the quantum theory, in terms of which one can efficiently perform the perturbative computations.
DOI: 10.1201/b15364-7