Path Integration in Quantum Mechanics

Janos Polonyi University of Strasbourg (Dated: February 12, 2021)

Contents

I. Trajectories in classical and and quantum mechanics 2

II. Brownian motion 3

IV. Direct calculation of the path integral 10 A. Free particle 10 B. Stationary phase (semiclassical) approximation 11 C. Quadratic potential 12

V. Matrix elements 14

VI. Expectation values 16

VII. Perturbation expansion 17

VIII. Propagation along fractal trajectories 18

IX. External electromagnetic ﬁeld 20

X. ˆIto integral 22

XI. Quantization rules in polar coordinates 23

A. Bracket formalism 25

B. Functional derivative 27

C. Representations of time dependence 27 2

1. Schr¨odinger representation 28 2. Heisenberg representation 28 3. Interaction representation 29 4. Schr¨odinger equation with time dependent Hamiltonian 29

I. TRAJECTORIES IN CLASSICAL AND AND QUANTUM MECHANICS

The formal structure of quantum mechanics prevents us to use our intuition in interpreting the basic equations. The path integral formalism offers an alternative where some ingredients of classical mechanics can be salvaged. The starting point of the mechanics is the concept of the state of motion, the set of informations which specifies the history of a point particle as the function of the time. The Newton equation is second order in the time derivative hence we need two data per degree of freedom to identify the time evolution, described by the trajectory, x(t). The Schrödinger equation is first order in the time derivative thus it is sufficient to specify the wave function at the initial time and the quantum mechanical state can be specified by the help of the coordinate alone. This is a well known problem, the Heisenberg uncertainty principle, ∆x∆p ~/2, forces us to use the coordinate ≥ or the momentum or a combination of the two to define the state of motion. The main victim of the restriction is the trajectory, x(t). In fact, had we known the trajectory of a particle by a continuous monitoring of its location then we would have access to the coordinate and velocity, ie. the momentum simultaneously. The path integral formalism, guessed by Dirac and worked out in detail by Feynman, offers an alternative way to describe the transition amplitudes of a particle in quantum mechanics in terms of trajectories. Naturally the trajectory of this formalism is not unique, we have actually an integration over trajectories. Imagine the propagation of a particle from the point S to D where a particle detector is placed in such a manner that a number of screens, each of them containing several small holes, are placed in between the source and the detector, c.f. Fig. 1. The particle propagates through the holes and the amplitude of detecting the particle, , is the sum over the possible ways of reaching the A detector. This is a sum over rectangles from the source to the detector. In the limit when the screens are placed closed to each others thus the particle traverses the next screen after a short time of flight and the size of a hole become small and close to each others a rectangle approaches 3

FIG. 1: The particle propagates from the source S to the location of the detector D through the holes of a system of screens. The amplitude of propagation is the sum of all possible ways of reaching the detector. a trajectory and the amplitude can be written as

= (path). (1) A A pathX This result reintroduces a part of classical physics in quantum mechanics and oﬀers a help to our intuition towards the understanding of quantum physics.

II. BROWNIAN MOTION

It is instructive to consider the problem of random walk where the path integral formalism arises in an intuitively clear and obvious manner. The probabilistic description of a classical particle is based on the probability density p(x,t) and the probability current j(x,t), satisfying the continuity equation

∂ p = ∇j. (2) t −

Fick’s equation relates the current to the inhomogeneity of the probability density

j = D∇p, (3) − in the absence of external forces, where D denotes the diﬀusion constant. The continuity equation allows us to write

∂tp = D∆p. (4) 4

This equation, called the diﬀusion or heat equation can formally be related to the Schr¨odinger equation,

~2 i~∂ ψ = ∆ψ (5) t −2m by the Wick rotation, ie. the analytic continuation to complex time, t i~t and the Sch ↔ − diff ~2 replacement D. 2m ↔ The Green function of the diﬀusion equation, (4), which corresponds to the initial condition p(x,t ) = δ(x x ) is called the Green function of eq. (4) and will be denoted by G(x,t ; x ,t ) i − i f i i for tf > ti. It is the conditional probability density that the particle is found at x at the time tf assuming that its location was x at time t : p(to from) = G(to; from). The solution of the i i ← diﬀusion equation which corresponds to the generic initial condition p(x,ti)= pi(x) can be written as

3 p(x,t)= d yG(x,t; y,ti)pi(y). (6) Z Te verify this claim we have to check two points:

1. Solution: The diﬀusion equation is linear hence this expression, a linear superposition of solutions is a solution, too.

2. Initial condition: It satisﬁes the desired initial condition,

3 p(x,ti) = d yG(x,ti; y,ti)pi(y) Z = d3yδ(x y)p (y) − i Z = pi(x). (7)

The conditional probability,

p(A B) p(A B)= ∩ , (8) | p(B) gives p(A B)= p(A B)p(B), and ∩ |

p(A B)= p(A B )p(B ) (9) ∩ | j j Xj B = B , B B = . The comparison of eqs. (6) and (9) where p(B) = 1 yields the interpretation ∪j j j∩ j ∅ of the Green function G(x,ti; xi,ti) as the conditional probability that the particle moves from xi at ti to x at t. 5

The expression (6) solves the diﬀusion equation for an arbitrary initial condition hence the equation

∂tG(x, t, y; ti)= D∆xG(x,t; y,ti) (10) follows for the Green function. It is easy to verify that

(x−y)2 1 4D(t −t ) G(x,t , y; ti)= e− f i (11) f [4πD(t t )]d/2 f − i satisﬁes eq. (10) in dimension d and the initial condition G(x, t, y; t)= δ(x y) as t t . − → i The particle must be somewhere at a given, ﬁxed intermediate time ti

3 3 p(x,t)= d zd yG(x, t, ; z,t′)G(z,t′; y,t0)pi(y). (12) Z

The expression (6) of the left hand side, valid for arbitrary pi(y) yields the Chapman-Kolmogorov equation,

3 G(x, t, xi; ti)= d zG(x, t, ; z,t′)G(z,t′; xi,t0). (13) Z By breaking the ﬁnite time of the propagation, t t into N +1 parts and applying the Chapman- − i Kolmogorov equation N-times one ﬁnds

3 3 G(x,t; xi,ti) = d z1 d zN G(x,t; zn,tn)G(zn,tn; zn 1,tn 1) · · · − − · · · Z G(z ,t ; x ,t )G(z ,t ; x ,t ) · · · 2 2 1 1 1 1 i i z −z 1 3 3 ∆t N ( n+1 n )2 = d z1 d zN e− 4D n=0 ∆t , (14) (4πD∆t)3(N+1)/2 · · · P Z where tn = ti +n∆t,∆t = 1/(N +1), z0 = xi and zN+1 = x. The right hand side can be considered as a summation over paths, made by piecewise linear functions which becomes an integral over paths in the continuum limit, N , and can formally be written as →∞ 1 t dt′x˙ 2 x x x − 4D ti G( ,t; i,ti)= D[ ]e R , (15) Z where the integration is over trajectories with initial and end points x(ti) = xi and x(t) = x, respectively and divergent normalization factor of the second line of eqs. (14) is included in the integral measure. Such an integration over trajectories is called Wiener process. A word of caution about the continuous notation: Almost all trajectory of the Wiener process is non-differentiable. In other word, the differentiable trajectories have vanishing weight in the Wiener integral in the limit N . The heuristic argument goes by inspecting the finite difference of → ∞ 6 trajectories with ∆t-independent weight, they must have ∆x = x x = √∆t according to n n+1 − n eq. (14), therefore v =∆x/∆t = ∆t 1/2 . The Wiener process is concentrated on fractals and O − the velocity,x ˙, appearing in the continuous notation of (15) must be handled symbolically, e.g. should be replaced by the discrete version, (14), as soon as it is used in a calculation.

III. PROPAGATOR

Let us consider a one dimensional non-relativistic particle described by the Hamiltonian

p2 H = + U(x) (16) 2m with [x,p]= i~ and introduce the propagator or transition amplitude

i Ht x e− ~ x (17) h f | | ii between coordinate eigenstates.

The amplitude (17) is a complicated function of the variables t, xi and xf . We simplify the problem of finding it by computing it first for short time when it takes a simpler form and by constructing the finite time transition amplitude from the short time one. This latter step is accomplished by writing

N i Ht i H∆t x e− ~ x = x e− ~ x (18) h f | | ii h f | | ii with ∆t = t/N and inserting a resolution of the identity,

11= dx x x , (19) | ih | Z between each operator,

N 1 i − i i i ~ Ht ~ H∆t ~ H∆t ~ H∆t xf e− xi = dxj xN e− xN 1 xN 1 e− xN 2 x1 e− x0 , (20) h | | i h | | − ih − | | − i···h | | i jY=1 Z where x0 = xi and xN = xf . This relation which holds for any N becomes a path integral as N . In fact, any trajectory between the given initial and ﬁnal point can be approximated by a →∞ piecewise constant function when the length of the time interval ∆t when the function is constant tends to zero. In order to turn the simple path integral expression (20) into something useful we need a simple approximation for the short time transition amplitudes. There are (N) of them multiplied O 7 together therefore it is enough to have N 1 = (∆t) accuracy in obtaining them. The ﬁrst O − O guess would be

i H∆t i x e− ~ x′ x 1 H∆t x′ h | | i ≈ h | − ~ | i i∆t x H x′ = x x′ 1 ~ h | | i h | i − x x′ ′ h | i i hx|H|x i ~ ∆t ′ x x′ e hx|x i (21) ≈ h | i but the problem is the orthogonality of the basis vectors x x = δ(x x ). In fact, the small h | ′i − ′ parameter in the expansion is ∆t/ x x which is diverging for x = x . To avoid this problem we h | ′i 6 ′ use two overlapping basis in an alternating manner. In case of continuous space the choice of the other, overlapping basis is rather natural. It will be a momentum basis, p with p q = q q . The | i | i | i corresponding resolution of the identity,

dp 11= p p , (22) 2π~| ih | Z inserted in Eqs. (21) yields

i H∆t dp i H∆t x e− ~ x′ = x e− ~ p p x′ h | | i 2π~h | | ih | i Z dp i x 1 H∆t p p x′ ≈ 2π~h | − ~ | ih | i Z dp i∆t x H p = x p p x′ 1 h | | i 2π~h | ih | i − ~ x p Z h | i dp i p(x x′) i ∆tH(p,x) e ~ − − ~ (23) ≈ 2π~ Z with

x H p p2 H(p,x)= h | | i = + U(x). (24) x p 2m h | i By replacing this expression into Eq. (20) we arrive at a path integral in phase space,

N 1 N i − dp i N xℓ−xℓ−1 ~ Ht k ~ ∆t [pℓ H(pℓ,xℓ)] xf e− xi = lim dxj e ℓ=1 ∆t − , (25) h | | i N 2π~ P →∞ jY=1 Z kY=1 Z with x0 = xi and xN = xf which can be written in a condensed, formal notation as

x(t)=x i f i ~ Ht ~ dτ[p(τ)˙x(τ) H(p(τ),x(τ))] xf e− xi = D[x] D[p]e − (26) h | | i R Zx(0)=xi Z by suppressing the regulator, ∆t. The integration over coordinate or momentum trajectories of ﬁxed or free initial and ﬁnal points, respectively. We shall see that the continuous notation is as 8 symbolic as for the Wiener measure, but notice here already that there is one more momentum integral than coordinate integration in eq. (25), preventing the quantum mechanical formalism to display canonical invariance which would follow only for canonical invariant integral measure,

j dxjdpj. Q TheR integrand in the exponent seems to be the Lagrangian of the analytical mechanics. However rather than relying on this classical analogy the momentum integral has to be performed in the quantum case. We start with Gauss’ formula,

∞ a x2 2π dxe− 2 = , (27) a Z−∞ r valid if for a> 0, yielding

2 ∞ a x2+bx 2π b dxe− 2 = e 2a (28) a Z−∞ r 2 after writing the exponent of the integrand in the form a (x b )2 + b and carrying out the − 2 − a 2a change of integration variable, x x + b . The integral → a

∞ i a x2+ibx dxe 2 (29) Z−∞ with real a can be calculated by analytic continuation by assuming Im a> 0. The correct Riemann- a sheet of the square root is chosen by requiring Re i > 0, giving rise to the Fresnel integral,

2 ∞ i a x2+ibx 2π i b +isign(Re a) π dxe 2 = e− 2a 4 (30) s a Z−∞ | | This result allows us to write the integral (25) as

N 1 i m − m i N m xℓ−xℓ−1 2 ~ Ht ~ ∆t [ ( ) U(xℓ)] xf e− xi = lim dxje ℓ=1 2 ∆t − (31) h | | i 2πi~∆t N 2πi~∆t P r →∞ j=1 r Y Z in coordinate space which reads in condensed notation

x(t)=xf i Ht i S[x] x e− ~ x = D[x]e ~ (32) h f | | ii Zx(0)=xi with

m S[x]= dτL(x(τ), x˙ (τ)), L = x˙ 2 U(x). (33) 2 − Z The expressions (26) and (32) are easy to memorize but are formal because the functional integration measure is deﬁned by a limiting procedure, spelled out in the more involved expressions (25) and (31). The integration over the momentum recovers the Lagrangian (33) from the Hamiltonian 9

(16). Such an agreement with the Legendre transformation of Classical Mechanics is restricted to the Gaussian integration, i.e. Hamiltonians of the form (16), which are quadratic in the momentum and contain the dependence in the coordinate as an additive term. We consider now few trivial but important generalizations of eq. (32). First seek the solution

ψ(t,x) of the Schr¨odinger equation, corresponding to the initial condition, ψ(ti,x) = ψi(x) by inserting a closing relation into

i S[x] ψ(t,x)= D[x]e ~ ψi(x(ti)) (35) Zx(t)=xf where the integration of the initial point is carried out with the weight, given by the wave function of the initial state. Another generalization consistes of the calculation of the matrix element

i Ht i Ht ψ e− ~ ψ = dx dx ψ∗(x ) x e− ~ x ψ (x ) h f | | ii i f f f h f | | ii i i Z i ~ S[x] = D[x]e ψf∗(x(tf ))ψi(x(ti)). (36) Z Finally, we generalize the results for a particle moving in a d-dimensional space. The steps leading to Eq. (25) can easily be repeated in this case leading to

N 1 N d i − d p i N xℓ−xℓ−1 ~ Ht d k ~ ∆t [pℓ H(pℓ,xℓ)] xf e− xi = lim d xj e ℓ=1 · ∆t − (37) h | | i N (2π)d P →∞ jY=1 Z kY=1 Z with p2 H = + U(x) (38) 2m which takes the form

x(t)=x i f i ~ Ht ~ dτ[p(τ)x˙ (τ) H(p(τ),x(τ))] xf e− xi = D[x] D[p]e − (39) h | | i R Zx(0)=xi Z in condensed notation. The Lagrangian path integral reads in d-dimensions as

N 1 d d x −x i m 2 − m 2 i N m ℓ ℓ−1 2 ~ Ht d ~ ∆t [ ( ) U(xℓ)] xf e− xi = lim d xje ℓ=1 2 ∆t − h | | i 2πi~∆t N 2πi~∆t P →∞ jY=1 Z x(t)=x f i = D[x]e ~ S[x] (40) Zx(0)=xi with m S[x]= dtL(x˙ , x), L = x˙ 2 U(x). (41) 2 − Z 10

IV. DIRECT CALCULATION OF THE PATH INTEGRAL

A. Free particle

Let us start with the path integral for a one dimensional free particle with ﬁnite N,

N 1 m − m im N 2 ~ (xℓ xℓ−1) G0(xf ,xi,t) = dxj e 2 ∆t ℓ=1 − 2πi~∆t  2πi~∆t  P r j=1 r Y Z N 1 N  m − = dxj f(xℓ xℓ 1, ∆t), (42) 2πi~∆t   − − r j=1 ℓ=1 Y Z Y   where x0 = xi and xN = xf , ∆t = T/N and

m im x2 f(x,t)= e 2~t . (43) 2πi~t r We shall calculate this expression by a successive integration. For this end let us consider the single integral

i a z2+ibz+ic dzf(x z,t )f(z y,t )= N dze 2 , (44) − 1 − 2 Z Z 2 2 m t1+t2 m yt1+xt2 m x y m with a = ~ , b = ~ , c = ~ ( + ), and N = ~ . The straightforward t1t2 − t1t2 2 t1 t2 2πi √t1t2 application of the Fresnel integral yields

2 2πi i b +ic dzf(x z,t )f(z y,t ) = N e− 2a − 1 − 2 a Z r 2 2 2 m (yt1+xt2) x y m i ~ + + = e 2 − t1t2(t1+t2) t1 t2 2πi~t r = f(x y,t + t ), (45) − 1 2 in other words, the integrand f(x,t) is self reproducing during the integration. This property can be used to successively integrate in (42) with the result

im 2 m (x xi) G (x ,x ,t)= f(x x ,t)= e 2~t f − . (46) 0 f i f − i 2πi~t r To check the result we calculate the free propagator in the operator formalism, too. The time evolution operator is diagonal in momentum space,

2 i Ht dp i p t e− ~ = p e− ~ 2m p , (47) 2π~| i h | Z 11 and the propagator, its matrix element in the coordinate basis, is

2 i Ht dp i p t x e− ~ x = x p e− ~ 2m p x h f | | ii 2π~h f | i h | ii Z i p2 i dp t+ p(x xi) = e− ~ 2m ~ f − 2π~ Z im 2 m (x xi) = e 2t~ f − . (48) 2πi~t r As a useful excercice let us calulate the spread of the wave packet

x2 e− 2∆x2 ψi(x)= , (49) ∆x√2 without using the momentum representation, p

2 1 m im (x y)2 y ψ(t,x)= dye 2t~ − − 2∆x2 (50) √ 2πi~t ∆x 2r Z what we write in the form p

a y2+by+c ψ(t,x)= N dye− 2 (51) Z with N = m , a = 1 i m , b = i m x and c = i m x2. The Gaussian intagral gives 2√2πi~t∆x ∆x2 − t~ − t~ 2t~ q 2 2π b +c ψ(t,x) = N e 2a a r 2 m 1 +i m t2~2 ∆x2 t~ 2 2 2 x +c 2π − 1 + m = N e ∆x4 t2~2 r a 2 2 3 4 2π m ∆x x2 i m ∆x x2+c = N e− 2(t2~2+m2∆x4) e− 2t~(t2~2+m2∆x4) , (52) r a a wave packet of width

∆x(t)= t2v2 +∆x2, (53) p ~ with v = m∆x .

B. Stationary phase (semiclassical) approximation

The semiclassical approximation of the path integral consists of the approximation by assuming that the action changes fast in units of ~. A simple integral

i I = dxe ~ S(x) (54) Z 12 can be approximated in the limit ~ 0 by restraining the domain of integration to the regions → where the exponent of the integrand changes slowly, x x , S (x ) = 0. Let us suppose that there ∼ 0 ′ 0 is a single point x0 where this equation is satisﬁed and write

i i ′ i 2 ′′ 3 ~ S(x0)+ ~ (x x0)S (x0)+ ~ (x x0) S (x0)+ ((x x0) ) I = dxe − 2 − O − Z i i 2 ′′ i 3 ~ S(x0) ~ y S (x0)+ ~ (y ) = e dye 2 O Z i 2π~ ′′ = e ~ S(x0) eisign(ReS (x0))(1 + ~2 ). (55) S (x ) O s| ′′ 0 | In case of several locally constant regions, S′(xj) = 0, j = 1 ...N we sum over thes regions,

N i 2π~ ′′ I = e ~ S(xj ) eisign(ReS (xj ))(1 + ~2 ). (56) S (x ) O j=1 s| ′′ j | X This approximation can easily be extended to the path integral,

x(t )=x f f i = D[x]e ~ S[x] (57) A Zx(ti)=xi where the dominant contribution comes from the domain of integration where the phase of the integrand changes the slowest manner with the trajectories, around the classical trajectory,

δS[x] = 0. (58) δx(t) x=x | cl By expanding the exponent around the classical trajectory we ﬁnd

2 x(tf )=xf i i δS[x] i ′ δ S[x] ′ ′ 3 ~ S[xcl]+ ~ dt[x(t) xcl(t)] + ~ dtdt [x(t) xcl(t)] ′ [x(t ) xcl(t )]+ ((x xcl) ) = D[x]e − δx(t) 2 − δx(t)δx(t ) − O − A R R Zx(ti)=xi y(t )=0 2 i f i ′ δ S[x] ′ i 3 ~ dtdt y(t) ′ y(t )+ ~ (y ) ~ S[xcl] 2 δx(t)δx(t ) O = e D[y]e R . (59) Zy(ti)=0 Since y √~ the y3 term can be neglected in the formal limit ~ 0 whee the path integral ∼ O → is reduced to the product of a phase factor, containing the classical action and a path integral of a quadratic action with vanishing initial and and points. This limit corresponds to the semiclassical limit when the initial and the ﬁnal point of the propagation, xi and xf , respectively, are held ﬁxed.

C. Quadratic potential

We consider a particle with the Lagrangian

m mω2(t) L = x˙ 2 x2, (60) 2 − 2 13 as the next example. The integral we seek in this case for ﬁnite N is

mω2 N+1 N i N+1 m 2 ℓ−1 2 m 2 ~ (yℓ yℓ−1) ∆t yℓ I = dy e ℓ=1 2∆t − − 2 −1 (61) N 2πi~∆t j P jY=1 Z with y0 = yN = 0, ∆t = T/N and ωℓ = ω(∆tℓ). We shall use the vector notation,

~y = (y0,y1,...,yℓ) for the trajectory and write

(N+1)/2 m N i ~yA ~y I = d ye 2 N , (62) N 2πi~∆t Z where

2 1 0 . . . 0 0 ω2 0 0 . . . 0 0 − 0  1 2 1 . . . 0 0   0 ω2 0 . . . 0 0  − − 1     m  0 1 2 . . . 0 0  m∆t  0 0 ω2 . . . 0 0   −   2  AN = ~  .  ~  .  . (63) ∆t  ..  −  ..             2   0 0 0 . . . 2 1  0 0 0 ... ωℓ 1 0   −   −     2  0 0 0 . . . 1 2   0 0 0 . . . 0 ωℓ   −        The matrix AN can be brought into a diagonal form by a suitable basis transformation and the Fresnel integral yields in the basis where it is diagonal

N+1 N m 2 2πi IN = ~ 2πi ∆t s λj jY=1 N m m = , (64) 2πi~∆t ~∆tλ r j=1 j Y r where λj denotes the eigenvalues. We have λj > 0 for suﬃcient small ∆t since the second derivative of the kinetic energy is a negative semi-deﬁnite operator. We write this expression as

m 1 IN = (65) 2πi~∆t ~∆t r det m AN q and introduce the notation

2 1 0 . . . 0 0 ω2 0 0 . . . 0 0 − 0  1 2 1 . . . 0 0   0 ω2 0 . . . 0 0  − − 1     ~∆t  0 1 2 . . . 0 0   0 0 ω2 . . . 0 0   −  2  2  DN = det AN = det  .  ∆t  .  . m  ..  −  ..             2   0 0 0 . . . 2 1  0 0 0 ... ωN 1 0   −   −     2   0 0 0 . . . 1 2   0 0 0 . . . 0 ωN   −       (66) 14

It is easy to verify the recursive relation

2 2 Dn+1 = (2 ∆t ωn+1)Dn Dn 1. (67) − − −

The corresponding initial conditions are D = 1 and D = 2 ∆t2ω2. We deﬁne in this manner a 0 1 − 0 function φ(n∆t)=∆tDn which satisfy the equation

φ¨(t)= ω2(t)φ(t), (68) − together with the initial conditions φ(ti) = 0, φ˙(ti) = 1. The solution of this initial condition problem, φ(t,ti) gives

y(tf )=0 i tf ~ t dtL(y(t),y˙(t)) lim IN = D[y]e i N R →∞ Zy(ti)=0 m = . (69) 2πi~φ(t ,t ) r f i This result allows us to ﬁnd the path integral with the quadratic Lagrangian, (60),

x(tf )=xf t t i f dtL(x(t),x˙(t)) m i f dtL(x (t),x˙ (t)) D[x]e ~ ti = e ~ ti cl cl , (70) R 2πi~φ(t ,t ) R Zx(ti)=xi r f i where the classical trajectory, xcl(t) solves the equation of motion

x¨ (t)= ω2(t)x(t), (71) cl − together with the boundary conditions xcl(ti)= xcl(tf ) = 0. As a simple example we consider the case of the harmonic oscillator, ω(t)= ω, where

tf mω S[x ]= dtL(x (t), x˙ (t)) = (x2 + x2) cos ωT 2x x , (72) cl cl cl 2sin ωT f i − i f Zti with T = t t and φ(t,t )= sin ωT , giving the exact result f − i i ω x(t )=x t 2 f f i f m 2 mω 2 i mω 2 2 ~ t dt 2 x˙ 2 x mω ~ [(x +x ) cos ωT 2xixf ] D[x]e i − = e 2 sin ωT f i − . (73) R h i 2πi~ sin ωT Zx(ti)=xi r

V. MATRIX ELEMENTS

The path integral representation of the transition amplitude generates the matrix elements of observables, too. The matrix elements of the kind

i H(t t′) i Ht′ x e− ~ − F (ˆx)e− ~ x (74) h f | | ii 15 can be obtained within this formalism in a natural manner where the hat denotes an operator. In fact, we have

i H(t t′) i Ht′ i H(t t′) i Ht′ x e− ~ − F (ˆx)e− ~ x = dx′ dx′ x e− ~ − x′ x′ F (x) x′ x′ e− ~ x h f | | ii 1 2h f | | 1i h 1| | 2ih 2| | ii Z δ(x′ x′ )F (x′ ) 1− 2 1 i ′ i ′ ~ H(t t ) | {z~ Ht } = dx′ x e− − x′ F (x′) x′ e− x h f | | i h | | ii Z ′ ′ x(t)=xf x(t )=x i dtL(x(t),x˙(t)) i dtL(x(t),x˙(t)) = dx′ D[x]e ~ F (x′) D[x]e ~ ′ ′ R R Z Zx(t )=x Zx(0)=xi x(t)=xf i dtL(x(t),x˙(t)) = D[x]e ~ F (x(t′) (75) R Zx(0)=xi where it was used in the last equation that the integration over trajectories from x(0) = xi to x(t′)= x′ and from x(t′)= x′ to x(t)= xf together with the integration over x′ is equivalent with the integration over trajectories from x(0) = xi to x(t) = xf . One may insert several operators into the time evolution at 0

x(t)=x i i i f i ~ H(t t2) ~ H(t2 t1) ~ Ht1 ~ dtL(x(t),x˙(t)) xf e− − F1(ˆx)e− − F2(ˆx)e− xi = D[x]e F1(x(t1)F2(x(t2). h | | i R Zx(0)=xi (76) It is useful to rewrite these equations in the Heisenberg representation introduced in Appendix C,

x(t)=x f i i ~ dtL(x(t),x˙(t)) ~ Ht D[x]e F (x(t′) = xf e− F (ˆxH (t′)) xi , R h | | i Zx(0)=xi x(t)=x f i i ~ dtL(x(t),x˙(t)) ~ Ht D[x]e F1(x(t1)F2(x(t2) = xf e− F1(ˆxH (t2))F2(ˆxH (t1)) xi . (77) R h | | i Zx(0)=xi The generalization of the latter equation for arbitrary order of the insertion gives

x(t)=x f i i ~ dtL(x(t),x˙(t)) ~ Ht D[x]e F1(x(t1)F2(x(t2)= xf e− T [F1(ˆxH (t2))F2(ˆxH (t1))] xi . (78) R h | | i Zx(0)=xi The insertion of a functional F [x(t)] of the the coordinates, taken at diﬀerent times is

x(t)=x f i i ~ dtL(x(t),x˙(t)) ~ Ht D[x]e F1[x(t)] = xf e− T [F (ˆxH (t)] xi . (79) R h | | i Zx(0)=xi The generalization of the latter result for analytic functionals n ∞ 1 F [x(t)] = dt f (t ,...,t )x(t ) x(t ), (80) n! j n 1 n 1 · · · j n=0 X jY=1 Z in terms of the generating functional

x(t)=x f i Z[j]= D[x]e ~ dt[L(x(t),x˙(t))+x(t)j(t)]. (81) R Zx(0)=xi 16 is

x(t)=x f i δ ~ dtL(x(t),x˙(t)) D[x]e F [x(t)] = F Z[j] j=0 R δ i j(t) | Zx(0)=xi " ~ # i Ht = x e− ~ T [F (ˆx (t)] x . (82) h f | H | ii

The proof follows by noting that the factor xn(t) is generated by acting with the diﬀerential ~ n n n operator ( i ) δ /δx(t) . Matrix elements of observables composed by the coordinate and the momentum can be calculated in a similar manner, by the use of the generating functional

N 1 N x x 2 − i∆t N ℓ− ℓ−1 pℓ dpk [pℓ U(xℓ)+xℓjℓ+pℓkℓ] Z[j, k] = lim dxj e ~ ℓ=1 ∆t − 2m − N 2π P →∞ jY=1 Z kY=1 Z x(t)=xf i dt[x(t)˙x(t) H(p(t),x(t))+x(t)j(t)+p(t)k(t)] = D[x]D[p]e ~ − , (83) R Zx(0)=xi which can be written after integrating out the momentum as

N 1 m − m N im 2 i∆t [ (xℓ xℓ−1+∆tkℓ) U(xℓ)+xℓjℓ] Z[j, k] = lim dxje ℓ=1 2∆t~ − − ~ 2πi~∆t N 2πi~∆t P r →∞ j=1 r Y Z i dτ[ m x˙ 2(τ)+ m k2(τ)+mx˙(τ)k(τ) U(x(τ))+x(τ)j(τ)] = D[x]e ~ 2 2 − . (84) R Z

The insertion of the mixed observable F [p(t),x(t′)] is found as

x(t)=x f i ~ δ ~ δ ~ dtL(x(t),x˙(t)) D[x]e F [x(t)] = F , Z[j, k] j=k=0 R i δj(t) i δk(t ) | Zx(0)=xi ′ i Ht = x e− ~ T [F [ˆx (t), pˆ (t′)] x . (85) h f | H H | ii

VI. EXPECTATION VALUES

The matrix elements of the type x A x considered so far are useful in calculating scattering h f | | iiH amplitudes. The expectation values of an observable A at time t can be written in the form

i Ht i Ht A(t) = x e ~ Ae− ~ x h i h i| | ii i i ~ Ht ~ Ht = Tr[e− ρ(ti)e A] = Tr[ρS(t)AS ]

i i ~ Ht ~ Ht = Tr[ρ(ti)e Ae− ] = Tr[ρH AH (t)] (86) where ρ = x x is the density matrix of the initial state. Note that the expressions (74) and (86) i | iih i| diﬀer more than in a phase factor when the initial state is not an eigenstate of the Hamiltonian. 17

The path integral expressions for (86) can easily be derived. First we give the path integral formulas for the density matrix for a given time t. There will be two path integrals, one for the bra and the other is for the ket of the initial density matrix,

x+(t)=x+ x−(t)=x− + + + ρ(x ,x−,t) = dxi dxi− D[x ] D[x−] + − Z Zx (0)=xi Zx (0)=xi i + + i − − ~ dtL(x (t),x˙ (t)) ~ dtL(x (t),x˙ (t)) + e − ρi(x ,x−) (87) R R i i giving the density matrix at time t. The expectation values (86) for A = F (x) can be obtained by means of the generating functional

x+(t)=x x−(t)=x + + + Z[j , j−] = dxdxi dxi− D[x ] D[x−] + − Z Zx (0)=xi Zx (0)=xi i + + − − + + − − ~ dt[L(x (t),x˙ (t)) L(x (t),x˙ (t))+x (t)j (t)+x (t)j (t)] + e − ρi(x ,x−) (88) R i i by inserting the operator in question into either time axis, ~ δ + F (x(t)) = Tr[ρH F (xH (t))] = F Z[j , j−] j±=0 h i i δj+(t) | ~ δ + = F Z[j , j−] j±=0. (89) i δj (t) | − The two lines in these equations are equivalent for Hermitian operators F (x). There is no diﬃ- culty in extending these formulas for observables containing the coordinate and the momentum at arbitrary time, ~ ~ δ δ + + F [x(t),p(t′)] = Tr[ρH T [xH (t),pH (t′)]] = F , Z[j , k , j−, k−] j±=k±=0 h i i δj+(t) i δk+(t ) | ′ ~ ~ δ δ + + = F , Z[j , k , j−, k−] j±=k±=0(90), i δj (t) i δk (t ) | − − ′ i Ht because the time evolution operator e− ~ is absent in (85) when used for the expectation value rather than the matrix element.

VII. PERTURBATION EXPANSION

The importance of the path integral integral formalism should be clear by now, it can be use to obtain matrix elements and expectation values. But its usefulness is not clear because the path integral has only been calculated for noninteracting particles. When the interaction, the non- Gaussian part of the action is weak compared to the quadratic part then one expects perturbation expansion be applicable. It is actually much more simple than in the operator formalism because 18 we face only c-numbers in the path integrals. By assuming U(x) 0 we have ≈

i i ′ m 2 ′ ′ ~ Ht ~ dt x˙ (t ) U(x(t )) xf e− xi = D[x]e 2 − (91) h | | i R Z n n n ∞ ( i) i ′ m 2 ′ ~ dt x˙ (t ) = − dtj D[x]e 2 U(x(tj) n!~n R n=0 X jY=1 Z Z jY=1 which can be written according as

n n n i ∞ ( i) i ′ ′ ~ Ht ~ dt H(t xf e− xi = − dtj xf T [ U(x(tj)e− ] xi . (92) h | | i n!~n h | R | i n=0 X jY=1 Z jY=1 These equations show that the propagation in the presence of a potential can be viewed as a sequence of interactions with the potential separated by free propagation.

VIII. PROPAGATION ALONG FRACTAL TRAJECTORIES

A classical particle moves along a trajectory with analytic dependence on the time as long as the potential is analytic. This is diﬀerent in Quantum Mechanics. The reason is that the spacial separation x y scales with the square root of the time of the propagation, √t in the free | − | propagator

~ t i ∂2 m im (x y)2 x e 2m t y = e 2~t − , (93) h | | i 2πi~t r yielding diverging velocity, x y /t 1/√t Another way to see this is to note that the typical | − | ≈ trajectories satisfy x y ∆t~/m in the path integral (31). In fact, the contributions of the | − | ≈ trajectories with (x y)2 ∆pt~/m are suppressed by the rapidly oscillating phase of the integrand − ≫ and the trajectories with (x y)2 ∆t~/m have small entropy. But the most detailed result comes − ≪ from Eq. (31) in the limit t when the dependence of the numerical values of the path integral →∞ on the ﬁnal point x is negligible. We can then shift the integration variables x x +x which f ℓ → ℓ ℓ+1 decouples them and give

∆t~ ∆x2 i 1 . (94) h i≈ m h i

This result expresses the fact that ﬁne the time resolution becomes longer the trajectory appears. The rescaling ∆t λ∆t induces the rescaling →

2 1 2 1/2 L = √∆x √∆x = λ− L (95) → √λ X X of the length of the trajectory, a scaling law characteristic of fractals. 19

We can now see better the formal feature of the expressions (26) and (32): the velocityx ˙ appearing in them is diverging and ill deﬁned! The quadratic terms p2 orx ˙ 2 alone in the exponents of Eqs. (32) or (26), respectively, would be enough to smoothen out the trajectories and to render x˙ ﬁnite by the oscillating phase of the integrand. But these quadratic expressions are multiplied by ∆t. This factor reduces the impact of the kinetic energy and we loose one power of ∆t in the denominator of x˙ 2 . h i The propagation along fractals is not a mathematical artifact, rather it represents a central part of Quantum Mechanics. Its suppression cancels Heisenberg commutation relation. We shall check this by calculating the matrix elements of the operator [x,p]

i t ~ dτH(τ) [x,p] = xf T [[p,x]e− 0 ] xi h i h | R | i i t ~ dτH(τ) = xf T [(xℓ+1pℓ+1 pℓ+1xℓ)e− 0 ] xi , (96) h | − R | i where the index ℓ corresponds to the time the commutator is inserted in the matrix element and the time ordering is used to arrive at the desired order of the coordinate and momentum operators. The infinitesimal propagator (23) shows that the first term on the right hand side of Eq. (96) is indeed the matrix element of xp. We find ~ 2 δ δ δ δ [x,p] = Z[j, k] j=k=0 (97) h i i δj δk − δj δk | ℓ+1 ℓ+1 ℓ ℓ+1 by means of the generating functional Eq. (84). One differentiation with respect to ikℓ/~ brings down the factor m(xℓ xℓ 1)/∆t as expected from classical mechanics and we find the familiar − − looking result x x x x [x,p] = x m ℓ+1 − ℓ x m ℓ+1 − ℓ h i h ℓ+1 ∆t − ℓ ∆t i m = (x x )2 ∆th ℓ+1 − ℓ i i~ 1 . (98) ≈ h i Note that a smearing of the singularity of the fractals, the replacement of the scaling law (94) by ∆t1+ǫ~ ∆x2 i 1 , (99) h i≈ m h i with ǫ> 0 leads to

[x,p] i~ 1 ∆tǫ 0, (100) h i≈ h i → and the loss of the canonical commutation relation. In other words, the scaling law, (94), is an essential part of quantum mechanics, its slightest weakening reduces quantum mechanics to classical physics. 20

IX. EXTERNAL ELECTROMAGNETIC FIELD

In the presence of an external vector potential A the path integral (40) is changed into

d N 1 d i m 2 − m 2 ~ Ht d xf e− xi = lim d xj h | | i 2πi~∆t N 2πi~∆t →∞ jY=1 Z x −x x −x x +x i ∆t N [ m ( ℓ ℓ−1 )2 U(x )+ e ℓ ℓ−1 A( ℓ ℓ−1 )] e ~ ℓ=1 2 ∆t − ℓ c ∆t · 2 P × x(t)=x f i = D[x]e ~ dτL(x(τ),x˙ (τ)) (101) R Zx(0)=xi where

m e L = x˙ 2 U(x)+ x˙ A(x). (102) 2 − c ·

The compact, cutoﬀ independent notation is even more misleading than in the absence of external vector potential due to the need of the mid-point prescription, appearing explicitly in the ﬁrst equation of (101). To see the origin of this unexpected complication we assume that the vector potential is actually evaluated at an intermediate point

x = (1 η)xℓ + ηxℓ 1 (103) − − in the Eq. (101) and follow an inverse argument in proving the path integral formulas. Let us denote the value of the path integral (101) by ψ(xf ,t) and ﬁnd its equation of motion. The propagator (17) reads for time dependent Hamiltonian as

i ~ dτH(τ) xf T [e− ] xi (104) h | R | i and the equation of motion should be the Schr¨odinger equation. The strategy is similar than the direct construction of the path integral: an inﬁnitesimal change of time. The increase t t +∆t → corresponds one more integration in the regulated path integral, therefore we have

3/2 m d ψ(x,t +∆t) = d x′ 2πi∆t~ Z im 2 i exp (x x′) ∆tU(x + η(x′ x)) 2~∆t − − ~ − ie + (x x′) A(x + η(x′ x)) ψ(x′,t). (105) c~ − · − 21

The integral is dominated by the contributions x x for small ∆t due to the rapidly oscillating ′ ≈ phase of the integrand and we make an expansion for small y = x x , − ′ m 3/2 imy2 i ie ψ(x,t +∆t) = ddy exp ∆tU + y A 2πi∆t~ 2~∆t − ~ c~ · Z i ieη + ∆tηy ∂U y y ∂ A + ~ · − c~ j k k j · · · 1 1 y ∂ + (y ∂)2 + ψ(x,t) (106) × − · 2 · · · · where the U = U(x) and A = A(x). The nest step is the expansion of the integrand,

3/2 im 2∆teη m d yj (δ ∂ Aj )y ψ(x,t +∆t) = d ye 2~∆t jk− mc k k 2πi∆t~ Z i∆t ie e2 i∆tη 1 U + y A (y A)2 + y ∂U + − ~ c~ · − 2c2~2 · ~ · · · · 1 1 y ∂ + (y ∂)2 + ψ(x,t) × − · 2 · · · · 3/2 im 2∆teη m d yj (δ ∂ Aj )y = d ye 2~∆t jk− mc k k 2πi∆t~ Z 1 i∆t ie ie 1 y ∂ + (y ∂)2 U + y A (y A)(y ∂) − · 2 · − ~ c~ · − c~ · · e2 i∆tη (y A)2 + y ∂U + ψ(x,t). (107) −2c2~2 · ~ · · · · Finally, the Gaussian integration can easily be carried out, ~ 1 i ∆t 1 i∆t ∆te 1 ψ(x,t +∆t) = 1+ ∂ B− ∂ U + A B− ∂ √ 2m · · − ~ mc · · detB 2 i∆te 1 A B− A + ψ(x,t), (108) −2mc2~ · · · · · where

2∆teη B = δ ∂ A . (109) jk jk − mc k j

The expansion of B in ∆t further simpliﬁes the result, ~ ∆teη i ∆t 1 i∆t ψ(x,t +∆t) = 1+ ∂ A 1+ ∂ B− ∂ U mc · 2m · · − ~ 2 ∆te 1 i∆te 1 + A B− ∂ A B− A + ψ(x,t) mc · · − 2mc2~ · · · · · ∆teη i~∆t i∆t ∆te = 1+ ∂ A + ∆ U + A ∂ mc · 2m − ~ mc · i∆te2 A A + ψ(x,t). (110) −2mc2~ · · · · 22

The equation of motion is

i~eη ~2 i~e e2 i~∂ ψ(x,t) = ∂ A ∆+ U + A ∂ + A2 ψ(x,t) (111) t mc · − 2m mc · 2mc2 1 ~ e 2 i~e(η 1 ) = ∂ A(x) + U(x)+ − 2 ∂ A(x) ψ(x,t) 2m i − c mc · " # where the cutoff is removed, ∆t 0. The lesson of this calculation is that contrary to the naive → expectation the η-dependence survives the removal of the cutoff and we must use the mid-point prescription, η = 1/2 in order to recover the standard Schrödinger equation. This unexpected effect, namely that the details at time scale ∆t remain visible at finite time after the limit ∆t 0 → has been taken is called quantum anomaly.

X. ˆITO INTEGRAL

The surprising persistence of the η-dependence found in the previous section is the result of the fractal nature of the typical trajectories in the path integral. We shall show this by identifying a characteristic feature of ordinary time integrals occurring within the path integral. The usual properties, such as the rule of change of variable, of a Riemann integral is usually derived by replacing the integral with a sum and by performing appropriate limit. These rules may change if the functions in question are not regular. Let us consider the following change of variable:

tf df(x) xf df(x) dtx˙(t) = dx = f(x ) f(x ), (112) dx dx f − i Zti Zxi valid for continuously differentiable functions x(t) and f(x). What happens if x(t) is a fractal, in particular, a typical trajectory in the path integral? To find the answer we perform the calculation at small but finite ∆t. We start with the safe identity

N f(xf ) f(xi)= [f(xj) f(xj 1)], (113) − − − Xj=1 where x0 = xi and xN = xf . In the next step we check the sensitivity of the right hand side on the choice of the point where the integrand is evaluated. For this end we introduce the η parameter by deﬁning the point of evaluation

(η) xj 1 = (1 η)xj + ηxj 1, (114) − − − 23

(η) the notation fj = f(xj ), ∆j = xj xj 1 and write for Eq. (113) − − N (η) 2 2 (η) (η) df(xj 1) η 2 d f(xj 1) − − f(xf ) f(xi) f(xj 1)+ η∆j + ∆j 2 − ≈ − dx 2 dx ! Xj=1 (η) 2 2 (η) (η) df(xj 1) (η 1) 2 d f(xj 1) f(x ) + (η 1)∆ − + − ∆ − − j 1 − j dx 2 j dx2 − ! N (η) 1 2 (η) = ∆jf ′(xj 1)+ η ∆j f ′′(xj 1) . (115) − − 2 − Xj=1 The scaling law, (94), yields xf df(x) 1 i~ xf d2f(x) f(x ) f(x )= dx + η dt , (116) f − i dx − 2 m dx2 Zxi Zxi a modification of partial integration rule. The functions are assumed to be sufficiently regular and differentiable in standard integral and differential calculus. But the transformation of integrals for fractals (94) requires to keep one order of magnitude more in the finite difference, ∆t or ∆x and the result is a modification of the usual rules, like (116).

XI. QUANTIZATION RULES IN POLAR COORDINATES

The fact that the path integral is dominated by nowhere differentiable, fractal trajectories requires the modification of certain rules of standard analysis. This explains the circumstance that the quantum mechanics does not display canonical invariance as its classical counterpart, in particular, its rules depend on the choice of coordinate system. We demonstrate this feature by working out the naive rules of quantization in polar coordinates. These rules for a free particle in the usual coordinate systems are the following: One starts with the Lagrangian m L = x˙ 2, (117) 0 2 defines the momentum ∂L p = , (118) ∂x˙ and construct the Hamiltonian,

H = xp˙ L , (119) 0 − 0 expressed in terms of the momentum p2 H = . (120) 0 2m 24

The canonical commutation relations,

[xj,pk]= i~δj,k, (121)

~ give rise to the representation pj = i ∂xj and the Hamiltonian ~2 H = 2, (122) 0 −2m∇ which possesses translational and rotational symmetry,

[H0, p] = [H0, L] = 0, (123) where the momentum p and angular momentum L generates translations and rotations, respectively. One uses the parametrization

r sin θ cos φ  x = r sin θ sin φ (124)   r cos θ   in polar coordinates where the free Lagrangian

m L˜ = (r ˙2 + r2θ˙2 + r2 sin2 θφ˙2) (125) 0 2 yields the momenta

δL p = 0 = mr,˙ r δr˙ δL0 2 pθ = = mr θ,˙ δθ˙ δL0 2 2 pφ = = mr sin θφ.˙ (126) δφ˙

The Hamiltonian is therefore of the form

H˜ = p r˙ + p θ˙ + p φ˙ L 0 r θ φ − p2 p2 p2 = r + θ + φ . (127) 2m 2mr2 2mr2 sin2 θ

The corresponding operators are determined by the canonical commutation relations,

[r, pr] = [θ,pθ] = [φ, pφ]= i~ (128) 25 with the solution ~ ∂ ~ ∂ ~ ∂ p = , p = , p = . (129) r i ∂r θ i ∂θ φ i ∂φ These operators lead to the Hamilton operator ~2 2 1 2 1 2 H˜0 = ∂ + ∂ + ∂ . (130) −2m r r2 θ sin2 θ φ It is not difficult to see that this operator does not possess the usual symmetries, namely [H˜ , p] = 0, 0 6 [H˜ , L] = 0. 0 6 The correct Hamiltonian is obtained by means of the Laplace-Beltrami operator, ~2 1 2 1 1 1 2 H0 = ∂r(r ∂r)+ ∂θ(sin θ∂θ)+ ∂ , (131) −2m r2 r2 sin θ sin2 θ φ which differs from H˜0, ~ 2 H = H˜ + i p + cot θp (132) 0 0 2m r r θ The difference, an (~) term, is called Îto potential because the derivation of the Hamilton func- O tion, defined by the exponent of the phase space path integral, produces the result (132) and shows that the Îto potential arises from the scaling law (94).

Appendix A: Bracket formalism

The bracket formalismis of Dirac is very well suited to the need of linear algebra and quantum mechanics in particular. It consists of thefollowing two steps:

1. The scalar products, (φ, ψ) and (φ, Aψ) where A is an operator are written as φ ψ and h | i φ A ψ , respectively. h | | i 2. The symbols bra, φ , and ket, ψ are used independently. (This is the main point, the h | | i previous one faciliates this use only.) The ket denotes a vector, an element of a linear space, ψ and the bra stands for a linear functional over the vector ﬁeld, φ : C. The | i∈H h | H → bras and kets form two equivalent linear spaces, namely there is an invertible linear map, connecting them if is a Hilbert space. H The advantages of the bracket formalism can be seen in the following rules:

1. Projector onto the vector ψ : | i ψ ψ P ψ = | ih | . (A1) | i ψ ψ h | i 26

2. Closing relation of a basis n : {| i}

11= n n . (A2) | ih | n X In the case of a continuous spectrum the sum is replaced by an integral.

3. Projection of a vector ψ onto a basis n : | i {| i}

ψ = 11 ψ = n n ψ . (A3) | i | i | ih | i n X 4. One formally deﬁnes the eigenstates of the coordinate and the momentum operators,x ˆ and pˆ, respectively, asx ˆ x = x x ,p ˆ p = p p and impose the closing relations, | i | i | i | i dp 11= dx x x = p p . (A4) | ih | 2π~| ih | Z Z The convention of the normalization of the state p , leading to the denominator in the | i second integral is motivated below by the Fourier theorem.

5. The wave function of the state ψ is deﬁned as ψ(x)= x ψ . The closing relation gives | i h | i

ψ(x)= x ψ = x 11 ψ = dy x y y ψ = dy x y ψ(y) (A5) h | i h | | i h | ih | i h | i Z Z implies x y = δ(x y). h | i −

6. The wave function of the image of the state ψ after the action of the operator A is | i

[Aψ](x)= x A ψ . (A6) h | | i

~ i eg. x pˆ p = x p = p x p ψ (x)= x p = ce ~ xp. h | | i i ∇h | i h | i → p h | i

7. Wave function in momentum space is given by ψ˜(p) = p ψ . The closing relation gives in h | i this case

dq ψ˜(p)= p ψ = p 11 ψ = p q q ψ , (A7) h | i h | | i 2π~h | ih | i Z leading to p q = 2π~δ(p q). h | i −

8. Fourier theorem for the wave function,

i px dp i px ψ˜(p)= dxe− ~ ψ(x), ψ(x)= e ~ ψ˜(p) (A8) 2π~ Z Z 27

can be written by means of the closing relations and c =1 as

i px ψ˜(p) = p ψ = p 11 ψ = dx p x x ψ = dxe− ~ ψ(x), h | i h | | i h | ih | i Z Z dp i ψ(x) = x ψ = x 11 ψ = dp x p p ψ = e ~ pxψ˜(p) (A9) h | i h | | i h | ih | i 2π~ Z Z The linear space, obtained by extending the original Hilbert space with the basis vectors x | i and p is called a rigged Hilbert space. | i

Appendix B: Functional derivative

The functionals are deﬁned by mens of lattice discretization. A continuous function f(t) deﬁned in the interval t

tf N F [f]= dtf(t)g(t)=∆t fjgj (B1) ti Z Xj=1 are deﬁned by

δnF [f] 1 ∂nF [f] = , (B2) δf(τ ) δf(τ ) ∆tn ∂f ∂f 1 · · · n 1 · · · n n where tj 1 τj

The generalization of the Taylor expansion for multi-variable functions,

∞ (ǫ ∂)n f(x + ǫ)= · f(x) (B4) n! nX=0 is the functional Taylor expansion reads as

n ∞ 1 tf δ F [x + ǫ]= dtǫ(t) F [x]. (B5) n! δx(t) n=0 ti X Z

Appendix C: Representations of time dependence

Sometime it is useful to make time-dependent basis transformation in representing the time dependence. One usually encounter the following three cases: 28

1. Schr¨odinger representation

The traditional way to represent the time dependence, generated by a time-independen Hamil- tonian H, is letting the states to follow the time-dependence prescribed by Schr¨odinger’s equation

i~∂ ψ(t) = H ψ(t) , (C1) t| iS | iS leading to

i (t ti)H ψ(t) = e− ~ − ψ(t ) . (C2) | iS | i iS

The time is treated as a number, without replacing it with an operator. Hence the time-dependence is trivial for the observables and we can restrict our attention the time-independent operators, i~∂tAS = 0. The diﬀerent representations will agree at the initial time, ti.

2. Heisenberg representation

The classical fields acquire non-trivial time-dependence in field theory. the quantum generalization, quantum field theory, contains field operators with non-trivial time-dependence. This makes is necessary to reformulate the time dependence in terms of opeartors. The simplest possibility is to perform a time-dependent basisi transformation in te Schrödingr picture which stops the states,

i (t ti)H Ψ(t) = e ~ − Ψ(t) . (C3) | iH | iS

The transformaiotn of the opeerators is inferred by requiring that the matrix elements are identical in the two representations,

i i i i (t ti)H (t ti)H (t ti)H (t ti)H ψ(t) A ψ(t) = ψ(t ) e ~ − A e− ~ − ψ(t ) = ψ(t ) e ~ − A e− ~ − ψ(t ) , h |S S| iS h i | S | i i h i | S | i i ψ(t) S ψ(t) S A(t)H h | | i (C4) | {z } | {z } | {z } which induces the transformation

i i (t ti)H (t ti)H AH (t)= e ~ − ASe− ~ − . (C5)

Therefore, the operators satisfy the Heisenberg equation of motion,

i~∂tAH (t) = [AH ,H], (C6) with the initial conditions AH (ti)= AS. 29

3. Interaction representation

The perturbation expansion can be applied for the Hamiltonian H = H0 + H1 where H0 and

H1 represent the free, easily diagonalizable dominant part and the small, complicated interaction part, respectively. The perturbation expansion for the Heisenberg equation is rather cumbersome a reasonable compromise between the two preceding representations to place the time dependence, generated by the simple part of the Hamiltonian into the operators and leave the complicated but supposedly small part of the time dependence for the states where the usual Rayleigh-Schr¨odinger perturbation expansion is relatively simple. For this end we deﬁne

i (t ti)H0 Ψ(t) = e ~ − Ψ(t) (C7) | ii | iS which induces the transformation

i i (t ti)H0 (t ti)H0 Ai(t)= e ~ − ASe− ~ − (C8) for the operators. The state vector satisﬁes the equation of motion

i (t ti)H0 i~∂ Ψ(t) = H Ψ(t) + e ~ − (H + H ) Ψ(t) t| ii − 0| ii 0 1 | iS i i i (t ti)H0 (t ti)H0 (t ti)H0 = H Ψ(t) + e ~ − (H + H )e− ~ − e ~ − Ψ(t) − 0| ii 0 1 | iS = H (t) Ψ(t) (C9) 1i | ii is indeed a Schr¨odinger equation involving the interaction only. The operators follow the free Heisenberg equation,

i~∂tAi(t) = [Ai,H0]. (C10)

4. Schr¨odinger equation with time dependent Hamiltonian

The interaction representation requires the solution of Schr¨odinger’s equation with time dependent Hamiltonian,

i~∂ Ψ(t) = H(t) Ψ(t) . (C11) t| i | i

To obtain it in a closed form one introduces the time-ordered product, a modiﬁed multiplication rule for operators depending on the time. For a chain of operators A (t ), , A (t ) the time 1 1 · · · n n 30 ordered product is deﬁned by acting with the operators in the order of ascending time values. For two operators we have

T [A(t )B(t )] = Θ(t t )A(t )B(t )+Θ(t t )B(t )A(t ), (C12) A B A − B A B B − A B A where

1 t> 1,  Θ(t)=  1 t = 0, (C13)  2  0 t< 0.   The solution of Schr¨odinger’s equation is 

Ψ(t) = U(t,t ) Ψ(t ) (C14) | i i | i i where

i t dt′H(t′) − ~ ti U(t,ti)= T [e R ]. (C15)

To prove this result it is suﬃcient to show that the time evolution operator U(t2,t1) satisﬁes the equation of motion

i~∂tU(t,ti)= H(t)U(t,ti). (C16)

This can easily be done by writing

i t dt′H(t′) ~ − ~ ti i ∂tU(t,ti) = i∂tT [e R ] i ∞ ( )n t t = i~∂ − ~ dt dt T [H(t ) H(t )], (C17) t n! 1 · · · n 1 · · · n n=0 ti ti X Z Z and noting that the derivative ∂t generates n-times the same integrand,

i n t t ∞ n( ~ ) i~∂tU(t,ti) = ~ − dt1 dtn 1T [H(t)H(t1) H(tn 1)] − − n! ti · · · ti · · · nX=0 Z Z i n 1 t t ∞ ( ~ ) − = H(t) − dt1 dtn 1T [H(t1) H(tn 1)] (n 1)! · · · − · · · − n=0 ti ti X − Z Z = H(t)U(t,ti). (C18)