Wick calculus ͒ Alexander Wurma Western New England College, Springﬁeld, Massachusetts 01119 Marcus Berg Max Planck Institute for Gravitational Physics, Albert Einstein Institute, D-14476 Potsdam, Germany ͑Received 19 February 2007; accepted 7 October 2007͒ In quantum ﬁeld theory the “normal ordering” of operators is routinely used, which amounts to shufﬂing all creation operators to the left. Potentially confusing is the occurrence in the literature of normal-ordered functions, sometimes called “Wick transforms.” In this paper, we introduce the reader to some basic results, ideas, and mathematical subtleties about normal ordering of operators and functions. The intended audience are instructors of quantum ﬁeld theory and researchers who are interested but not specialists in mathematical physics. © 2008 American Association of Physics Teachers. ͓DOI: 10.1119/1.2805232͔
I. INTRODUCTION polynomials. Finally, we provide a brief example of how the Wick transform can be utilized in a physical application. Normal ordering was introduced in quantum ﬁeld theory by Wick in 1950 to avoid some inﬁnities in the vacuum expectation values of ﬁeld operators expressed in terms of II. WICK OPERATOR ORDERING creation and annihilation operators.1 The simplest example of such inﬁnities can be discussed starting from only nonrel- A. Simple harmonic oscillator ativistic quantum mechanics and the simple harmonic oscil- The Hamiltonian operator for the simple harmonic oscil- lator; an inﬁnite number of harmonic oscillators make up a lator in nonrelativistic quantum mechanics has the form: free quantum ﬁeld. ͑The reader who wishes a quick reminder 1 ͑ 2 2͒ ͑ ͒ of some basic quantum ﬁeld-theoretical concepts may ﬁnd H = 2 P + Q , 1 comfort in Appendix A͒. where we have, as usual, hidden Planck’s constant ប, the In addition to the usual concept of quantizing by promot- ing ﬁelds to operators, modern quantum ﬁeld theory uses mass m, and the angular frequency in the deﬁnitions of the functional integrals as basic objects. In the functional inte- dimensionless momentum and position operators, gral formalism we calculate physical quantities such as scat- 1 m tering cross sections and decay constants by integrating over P ϵ pˆ, Q ϵ ͱ qˆ , ͑2͒ ͱបm ប some polynomials in the ﬁelds and their derivatives ͑see, for example, Refs. 2 and 3͒. In the examples considered here, the so that ﬁelds we want to integrate are equivalent to functions on ͓P Q͔ i ͑ ͒ spacetime. As mentioned, a ﬁrst step toward removing inﬁni- , =− . 3 ties and giving mathematical meaning to calculations in the If we deﬁne the creation and annihilation operators a† and a operator formalism is normal ordering. The analog of this by ordering for functional integrals is called the “Wick trans- 1 form”. a† = ͑Q − iP͒, ͑4͒ In this paper we introduce the reader to an interesting con- ͱ2 nection, which is usually not discussed in introductory treat- ments of quantum ﬁeld theory, among normal ordering, Wick 1 a = ͑Q + iP͒, ͑5͒ transforms, and Hermite polynomials. We will also be con- ͱ2 cerned with some basic questions that are usually glossed over, i.e., what does the functional integral really mean? Al- so that though a complete answer is not known and beyond the ͓a,a†͔ =1, ͑6͒ scope of this paper, we intend to give some ﬂavor of the ﬁrst steps toward addressing this question and how the Wick we ﬁnd that transform has been put to work in this regard. H = 1 ͑P2 + Q2͒ = 1 ͑a†a + aa†͒ = a†a + 1 . ͑7͒ The paper is organized as follows: We will use the har- 2 2 2 monic oscillator to exhibit the connection between Wick- The eigenvalues of this Hamiltonian operator ͑see for ex- ordered polynomials and the familiar Hermite polynomials. ample, Ref. 4͒ satisfy the sequence: Then we turn to Wick transforms in the functional integral 1 ͑ ͒ formalism of ﬁeld theory, where we show that there is again En = n + 2 , n = 0,1,2, ... . 8 a connection with Hermite polynomials. Several approaches In particular, the ground state energy ͑or zero-point energy͒, to Wick transforms are compared and shown to be equiva- which is the lowest eigenvalue of the Hamiltonian, is non- lent. In passing, we observe how the standard quantum ﬁeld zero: theory result known as “Wick’s theorem” follows directly in ͉ ͘ 1 ͉ ͘ ͑ ͒ this framework from well-known properties of the Hermite H 0 = 2 0 . 9
65 Am. J. Phys. 76 ͑1͒, January 2008 http://aapt.org/ajp © 2008 American Association of Physics Teachers 65 This zero-point energy has observable physical conse- q4 =:q4:+6:q2:+3. ͑14c͒ quences. As an illustration, it is possible to measure the zero- point motion of the atoms in a crystal by studying the dis- Because we can recursively replace normal-ordered terms on persion of light in the crystal. Classical theory predicts that the right by expressions on the left that are not normal- ͑ 2 2 ͒ the oscillations of the atoms in the crystal, and therefore also ordered for example :q : can be replaced by q −1 ,wecan dispersion effects, cease to exist when the temperature is also invert these relations: near absolute zero. However, experiments demonstrate that 2 2 ͑ ͒ ͑ ͒ :q :=q −1=He2 q , 15a the dispersion of light reaches a ﬁnite nonzero value at very low temperatures. 3 3 ͑ ͒ ͑ ͒ :q :=q −3q =He3 q , 15b In quantum ﬁeld theory a free scalar ﬁeld can be viewed as an inﬁnite collection of harmonic oscillators as described in :q4:=q4 −6q2 +3=He ͑q͒, ͑15c͒ Appendix A. If we proceed as before, each oscillator will 4 ͑ ͒ give a contribution to the zero-point energy, resulting in an where the polynomials Hen q are a scaled version of the inﬁnite energy, which seems like it could be a problem. more familiar form of the Hermite polynomials Hn: One way to remedy the situation is to deﬁne the ground ͑ ͒ −n/2 ͑ /ͱ ͒ ͑ ͒ state as a state of zero energy, which can be achieved by Hen x =2 Hn x 2 . 16 redeﬁning the Hamiltonian. We subtract the contribution of In some of the mathematical physics literature, the Hen are the ground state and deﬁne a Wick-ordered ͑or normal- often just called Hn. Some of the many useful properties are ordered͒ Hamiltonian, denoted by putting a colon on each collected in Appendix B for easy reference ͑a more complete side, by collection is given in Ref. 5, for example͒. :H:=:1 ͑a†a + aa†͒: ϵ 1 ͑a†a + aa†͒ − 1 ͗0͉a†a + aa†͉0͘ Because of the relation between operator Wick ordering 2 2 2 and Hermite polynomials, the literature sometimes deﬁnes ϵ a†a. ͑10͒ “Wick ordering” in terms of Hermite polynomials: n ϵ ͑ ͒ ͑ ͒ In this example the deﬁnition of Wick ordering can be :q : Hen q . 17 thought of as a redeﬁnition of the ground state of the har- Although q is an operator composed of noncommuting op- monic oscillator. † In the last equality in Eq. ͑10͒ we see that all creation erators a and a , this alternative deﬁnition naturally general- operators end up on the left. A general prescription for Wick izes to Wick ordering of functions. We will explore this idea ordering in quantum ﬁeld theory in a creation/annihilation in the next section. operator formalism is “Permute all the a† and a, treating One reason that the connection to Hermite polynomials is them as if they commute, so that in the end all a† are to the not mentioned in the standard quantum ﬁeld theory literature is the fact ͑which was also Wick’s motivation͒ that the left of all a.” The resulting expression is the same: normal-ordered part is precisely the part that will vanish :H:=a†a. ͑11͒ when we take the vacuum expectation value. The traditional way to deﬁne normal-ordering, the one given at the end of Sec. II A ͑“put a† to the left of a”͒, yields for powers of q: B. Wick ordering and Hermite polynomials n n The ﬁrst connection between Wick ordering and Hermite :qn:=͚ ͩ ͪ͑a†͒n−iai, ͑18͒ polynomials arises when we study powers of the ͑dimension- i=1 i less͒ position operator Q. For the harmonic oscillator the eigenvalue of Q2 gives the variance of the oscillator from which vanishes for any nonzero power n when applied to the vacuum state ͉0͘. rest. We have Q=͑a† +a͒/ͱ2, but to avoid cluttering the ͱ In other words, because we know that normal-ordered equations with factors of 2, we will study powers of just terms vanish upon taking the vacuum expectation value, we † ͑a +a͒: may not be interested in their precise form. However, if the ͑ ͑a† + a͒2 = a†2 + a†a + aa† + a2, ͑12a͒ expectation value is not taken in vacuum for example, in a particle-scattering experiment͒, this part does not vanish in =a†2 +2a†a + a2 + ͓a,a†͔, ͑12b͒ general, and there are many instances where the actual normal-ordered expression itself is the one of interest. =:͑a† + a͒2:+͓a,a†͔, ͑12c͒ III. FUNCTIONAL INTEGRALS AND THE WICK =:͑a† + a͒2:+1 ͓by Eq. ͑10͔͒. ͑12d͒ TRANSFORM If we arrange terms in a similar way for higher powers of ͑a† +a͒, we ﬁnd: Most contemporary courses on quantum ﬁeld theory dis- cuss functional integrals ͑sometimes called path integrals; † 3 † 3 † ͑a + a͒ =:͑a + a͒ :+3͑a + a͒, ͑13a͒ however, only in nonrelativistic quantum mechanics do we really integrate over paths͒. In a functional-integral setting, ͑a† + a͒4 =:͑a† + a͒4:+6:͑a† + a͒2:+3. ͑13b͒ the counterpart of the Wick ordering in the operator formal- We can summarize the results using the notation a† +a=q, ism is the Wick transform. This transform applies to func- tions and functionals. It can, like its quantum-mechanics q2 =:q2:+1, ͑14a͒ counterpart ͓Eq. ͑17͔͒, be deﬁned by means of Hermite poly- nomials. But ﬁrst, we brieﬂy skip ahead and explain why q3 =:q3:+3:q:, ͑14b͒ such a transform will prove to be useful.
66 Am. J. Phys., Vol. 76, No. 1, January 2008 Alexander Wurm and Marcus Berg 66 A. Integration over products of ﬁelds In general, the covariance is a positive continuous nonde- generate bilinear form on the space of test functions. A In the functional integral formalism, physical quantities simple example is the covariance for a free Klein–Gordon such as scattering cross sections and decay constants are cal- field, culated by integrating over some polynomial in the ﬁelds and their derivatives. The algorithmic craft of such calculations is ͑ ͒ ͵ 4 4 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ described in textbooks such as those of Peskin and C f,g = d xd yf x DF x − y g y , 22 Schroeder2 and Ryder.3 Although there are examples of ͑ physical effects that can be studied with functional integral where DF is the textbook Feynman propagator see Ref. 2 for methods but not with ordinary canonical quantization,6 in the example͒. In the following we will often encounter the co- scope of this paper we can only give examples of some re- variance at coincident test functions, here denoted as C͑f , f͒. sults that can be derived more quickly or transparently using To get to the point, a Gaussian measure d C is deﬁned by functional integrals. its covariance C as The polynomials we will consider in this section are those ͑ of Euclidean ﬁelds ﬁelds deﬁned on four-dimensional Eu- ͵ d ͑⌽͒exp͑− i͗⌽, f͒͘ = exp͓− 1 C͑f, f͔͒, ͑23͒ 4͒ C 2 clidean space R . Similar formalisms exist for Minkowski Y ﬁelds ͑ﬁelds deﬁned on spacetime͒ with minor changes in the equations ͑see for example, Ref. 2͒. Because functional inte- over a space Y of distributions ⌽. For comparison, the usual Gaussian measure on Rd is de- grals over Minkowski ﬁelds are less mathematically devel- 9 oped than integrals over Euclidean ﬁelds, we shall restrict ﬁned by, our attention to Wick transforms of functions and functionals 1 ddx of Euclidean ﬁelds—primarily polynomials and exponen- ͵ e−͑1/2͒Q͑x͒e−i͗xЈ,x͘ = e−͑1/2͒W͑xЈ͒, ͑24͒ ͑ ͒d/2 ͑ ͒1/2 tials. The Wick transform, like the Hermite polynomials, has 2 Rd det D orthogonality properties that turn out to be useful in quantum d d ͗xЈ x͘ d xЈx x ﬁeld theory. First, we have to introduce a few mathematical where , R = is the duality in R with R and Ј ͑ ͒ d concepts. x Rd, Q x is a quadratic form on R , ͑ ͒ ͗ ͘ ͑ ͒ Q x = Dx x = Dx,x Rd, 25 1. Gaussian measures ͑ ͒ and W xЈ is a quadratic form on Rd Here, our aim is to specify the notation and to brieﬂy W͑xЈ͒ = xЈC xЈ = ͗xЈ,CxЈ͘ d, ͑26͒ remind the reader how to integrate over Euclidean ﬁelds, R without going into too much detail. The standard mathemati- such that cal framework to perform such integration is the theory of Gaussian measures in Euclidean ﬁeld theory.7,8 DC = CD =1. ͑27͒ As a ﬁrst try, we might deﬁne a ﬁeld in the functional d ͓ ͑ ͔͒ 4 In the more familiar R case Eq. 24 , the combination of integral as a function on R . Fields in the functional inte- the standard measure and kinetic term corresponds to the gral might seem like functions at ﬁrst glance, but can pro- measure d C we introduced earlier. That is, duce divergences that cannot, for instance, be multiplied in ddx/͑det D͒1/2e−1/2Q͑x͒ is analogous to d ͑⌽͒. The explicit the way that functions can. A more useful way to regard a C d ddx/͑ D͒1/2 e−1/2Q͑x͒ quantum ﬁeld in the functional integral formalism is as a separation of C into det and will turn distribution ⌽ acting on a space of test functions f: out to be unnecessary for our discussion. By deﬁning the ͑ ͒ measure d C through Eq. 23 , we have not even speciﬁed ⌽͑f͒ϵ͗⌽, f͘, ͑19͒ what such a separation would mean. where the bracket ͗,͘ denotes duality, that is, ⌽ is such that it The covariance at incident points can be expressed as the following integral, obtained by expanding Eq. ͑23͒: yields a number when applied to a smooth test function f.In many situations the distribution ⌽ is equivalent to a function ͑ ͒ ͑⌽͒͗⌽ ͘2 ͑ ͒ , which means this number is the ordinary integral: C f, f = ͵ d C , f . 28 ͑ ͒ ͗⌽, f͘ = ͵ d4x͑x͒f͑x͒. ͑20͒ The integral on the left-hand side of Eq. 23 is the generat- R4 ing function of the Gaussian measure; let us denote this in- tegral by Z͑f͒. By successively expanding Eq. ͑23͒, the nth A familiar example of a distribution is the Dirac distribution ⌽ ␦ moment of the Gaussian measure can be compactly written = , for which we have as ⌽͑f͒ = ͗␦, f͘ = f͑0͒. ͑21͒ d n ͯ͵ d ͑⌽͒͗⌽, f͘n = ͩ− i ͪ Z͑f͒ͯ Just like a function, ⌽ in general belongs to an infinite- C d =0 dimensional space. To be able to integrate over this space n/2 ͓not to be confused with the integral in Eq. ͑20͒, which is ͑n −1͒ !!C͑f, f͒ n even = ͭ an ordinary integral over spacetime͔ we need a measure, 0 n odd, 4 some generalization of the familiar d x in the ordinary ͑29͒ integral in Eq. ͑20͒. To this end, we need to introduce the ͑ ͒͑ ͒ covariance. Given the action of the theory, we readily where n!!=n n−2 n−4 ¯ is the semifactorial. For conve- calculate the classical equations of motion, extract the dif- nience, we introduce the following notation for the average ferential operator D, and invert it to find the covariance C. with respect to the Gaussian measure C:
67 Am. J. Phys., Vol. 76, No. 1, January 2008 Alexander Wurm and Marcus Berg 67 :exp͑␣⌽͑f͒͒: ϵ 1+␣:⌽͑f͒: + 1 ␣2:⌽͑f͒2: + ... , ͗ ͓⌽͑ ͔͒͘ ϵ ͵ ͑⌽͒ ͓⌽͑ ͔͒ ͑ ͒ C C 2 C F f d C F f . 30 C ͑35͒
Note the difference between the brackets ͗͘ used for the where normal ordering is deﬁned by Eq. ͑33͒. C average and the brackets ͗,͘ used for duality. Problem 1. Show that exp͑␣⌽͑f͒͒ ͑␣⌽͑ ͒͒ ͑ ͒ :exp f :C = . 36 ͗exp͑␣⌽͑f͒͒͘ C 2. Wick transforms: Deﬁnitions Solution. We can evaluate the right-hand side of Eq. ͑36͒ by We can use this set of deﬁnitions to deﬁne a Wick trans- expanding the numerator and denominator in a power series form of functionals of ﬁelds. Our goal is to provide some and dividing one power series by the other. idea of how to address the difﬁcult problem of making sense ϱ ͚ b xk ϱ out of products of distributions and integrals of such prod- k=0 k 1 k ͑ ͒ ϱ = ͚ ckx , 37 ucts. These products are ubiquitous in quantum ﬁeld theory, k ͚ a x a0 k=0 although their exact meaning is not usually discussed in in- k=0 k troductory textbooks. To simplify the calculations, we deﬁne 1 n n n where c + ͚ c a −b =0. A comparison of the resulting the Wick transform of a power ⌽͑f͒ ϵ͗⌽ , f͘ so as to sat- n a0 k=1 n−k k n isfy an orthogonality property with respect to Gaussian inte- series, term by term, to the power series expansion of the ͑ ͒ gration. We recall the orthogonality properties of Hermite left-hand side proves Eq. 36 . ͑ ͒ ͑ ͒ polynomials ͑Appendix B͒ and the deﬁnition ͓Eq. ͑17͔͒ and Problem 2. Show the equivalence of Eqs. 33 and 31 . deﬁne the Wick transform in terms of Hermite polynomials: Solution. We can explicitly calculate the denominator in Eq. ͑36͒: ⌽͑f͒ ⌽͑ ͒n ͑ ͒n/2 ͩ ͪ ͑ ͒ : f :C = C f, f Hen ͱ . 31 ͗ ͓␣⌽͑ ͔͒͘ ͵ ͑⌽͒ ͑␣⌽͑ ͒͒ ͑ ͒ exp f = d exp f C f, f C C Note that this deﬁnition depends on the covariance C, and ␣n ͵ ͑⌽͒ ⌽͑ ͒n ͑ ͒ that there is no analogous dependence in the analogous = d C ͚ f 38a harmonic-oscillator deﬁnition ͓Eq. ͑17͔͒. n n! The orthogonality of two Wick-transformed polynomials is expressed by ␣2n ͵ ͑⌽͒⌽͑ ͒2n ͑ ͒ =͚ d C f , 38b n n! ͵ ͑⌽͒ ⌽͑ ͒n ⌽͑ ͒m ␦ ͑͗⌽͑ ͒⌽͑ ͒͘ ͒n d : f : : g : = n ! f g . C C C m,n C ͓by Eq. ͑29͔͒ ͑32͒ 1 An entertaining problem is to prove Eq. ͑32͒, which we will =expͫ ␣2C͑f, f͒ͬ . ͓by Eq. ͑28͔͒ do in Sec. III A 3 ͑paragraph 2͒. 2 The Wick transform can also be deﬁned recursively by the ͑38c͒ following equations:10 Thus, from Eq. ͑36͒ we ﬁnd :⌽͑f͒0: =1, ͑33a͒ C ͓␣⌽͑ ͔͒ ͑␣⌽͑ ͒ 1 ␣2 ͑ ͒͒ ͑ ͒ :exp f :C = exp f − 2 C f, f . 39 ␦ If we multiply the power series expansions11 of exp͓␣⌽͑f͔͒ :⌽͑f͒n: = n:⌽͑f͒n−1: , n =1,2, ... , ͑33b͒ ␦⌽ C C and exp͓1/2␣2C͑f , f͔͒ and compare the result term by term to the series expansion of the left-hand side of Eq. ͑39͒,we obtain ͵ ͑⌽͒ ⌽͑ ͒n ͑ ͒ d C : f :C =0, n =1,2, ... , 33c ͓ n ͔ 2 n! 1 m :⌽͑f͒n: = ͚ ⌽͑f͒n−2mͩ− C͑f, f͒ͪ . where the functional derivative with respect to a distribution C m ! ͑n −2m͒! 2 is m=0 ͑40͒ ␦ ⌽͑f͒ = f . ͑34͒ ͑ ͒ ␦⌽ We rewrite Eq. 40 as ͓ n ͔ Let us check that the Wick transform :⌽͑f͒n: deﬁned by Eq. 2 n! C :⌽͑f͒n: = C͑f, f͒n/2 ͚ ͑−1͒m ͑33͒ is the same as the Wick transform given in terms of C m ͑ ͒ m=0 2 m ! n −2m ! Hermite polynomials in Eq. ͑31͒. This equivalence is to be expected, because the Hermite polynomials satisfy similar ⌽͑f͒ n−2m ϫͩ ͪ , ͑41͒ recursion relations, but it is a useful problem to check that it ͱC͑f, f͒ works. To begin, we establish a property of Wick exponen- tials. use the expression ͑B1͒ for the deﬁning series of the Hermite ͑␣⌽͑ ͒͒ Let :exp f :C be the formal series: polynomials given in Appendix B and recover Eq. ͑31͒.
68 Am. J. Phys., Vol. 76, No. 1, January 2008 Alexander Wurm and Marcus Berg 68 3. Wick transforms: Properties These latter multifield expressions reproduce, within this functional framework, what is usually referred to as Wick’s Many properties of Wick-ordered polynomials can be con- theorem in the creation/annihilation-operator formalism. In veniently derived using the formal exponential series. For this formalism it takes some effort to show this theorem; simplicity, we assume that all physical quantities that we here we find it somewhat easier by relying on familiar prop- ͑ ͒ wish to calculate for example, scattering cross sections are erties of the Hermite polynomials. written with normalization factors of 1/C͑f ,g͒, which in ef- fect lets us set the coincident-point covariance to unity: C͑f , f͒=1. The covariance can be restored by comparison with Eq. ͑41͒. The properties in which we are interested are B. Wick transforms and functional Laplacians useful problems: Problem 3. Show that :exp͑⌽͑f͒+⌽͑g͒͒: We can also deﬁne Wick transforms of functions by the C following exponential operator expression, which is conve- =exp͑−͗⌽͑f͒⌽͑g͒͘ ͒:exp͑⌽͑f͒͒: :exp͑⌽͑g͒͒: . C C C nient in many cases ͑for example in two-dimensional quan- Solution. tum ﬁeld theory settings, such as in Refs. 12 and 13͒:
:exp͑⌽͑f͒͒: :exp͑⌽͑g͒͒: , ͑42a͒ 1 ⌬ C C n͑ ͒ ϵ − Cn͑ ͒ ͑ ͒ : x :C e 2 x , 47
1 2 2 =exp͑⌽͑f͒ + ⌽͑g͒͒exp͑− ͓͗⌽͑f͒ ͘ + ͗⌽͑g͒ ͘ ͔͒, where the functional Laplacian is deﬁned by 2 C C ͑42b͒ ␦ ␦ ⌬ = ͵ d4xd4xЈC͑x,xЈ͒ , ͑48͒ C ␦͑x͒ ␦͑xЈ͒ =:exp͑⌽͑f͒ + ⌽͑g͒͒: exp͑͗⌽͑f͒⌽͑g͒͘ ͒, ͑42c͒ C C which, again, depends on the covariance C. ͑ ͒ where we have used Eq. ͑39͒ in the ﬁrst equality, and, after Instead of proving Eq. 47 , which is straightforward, we ͑ ͒ just illustrate the equivalence of deﬁnition ͑47͒ and deﬁnition completing the square in the second factor of Eq. 42b , ͑ ͒ 4 again in the second equality. Dividing both sides of Eq. ͑42͒ 31 for the common example of a power for which the by the second factor in Eq. ͑42c͒ completes the proof. deﬁnition becomes ͗ ⌽͑ ͒n ⌽͑ ͒m ͘ Problem 4. Show that : f :C : g :C : 1 ␦ ␦ C 4͑ ͒ ͩ 4 4 Ј ͑ Ј͒ ͪ4͑ ͒ n : y : = exp − ͵ d xd x C x,x y . =␦ n!͗⌽͑f͒⌽͑g͒͘ C ␦͑ ͒ ␦͑ ͒ nm C 2 x xЈ Solution . If we take the expectation value of both sides of ͑49͒ Eq. ͑42͒,weﬁnd ͗ ͑⌽͑ ͒͒ ͑⌽͑ ͒͒ ͘ ͓͗⌽͑ ͒⌽͑ ͒͘ ͔ We expand the exponential and ﬁnd :exp f :C:exp g :C = exp f g C C 1 ͑43͒ :4͑y͒: = 4 + ͩ− ͪͩ͵ d4xd4xЈC͑x,xЈ͒ C 2 ͑ ͒ using Eq. 33 . Expanding the exponentials on both sides and ␦ ␦ comparing term by term completes the proof. ϫ ͪ4͑y͒ ⌽͑ ͒n+1 ⌽͑ ͒n−1 ␦͑ ͒ ␦͑ Ј͒ Problem 5. Show that : f :C =n: f :C x x ⌽͑ ͒ ⌽͑ ͒n − f : f :C. 1 1 2 Solution. This recursive relationship is a consequence of + ͩ− ͪ ͩ͵ d4xd4xЈC͑x,xЈ͒ the equivalence of Wick-ordered functions and Hermite 2! 2 polynomials. The expression follows from the recursion re- ␦ ␦ 2 lation for Hermite polynomials given in Appendix B. ϫ ͪ 4͑y͒. ͑50͒ ␦͑x͒ ␦͑xЈ͒ Problem 6. The deﬁnition of Wick transforms given in Eq. ͑33͒ can be generalized to several ﬁelds in a straightforward All higher terms in the expansion are zero. We can now manner. We quote here some results without proof ͑details evaluate each term separately. The second term is the integral can be found in Ref. 10͒. The reader may ﬁnd it interesting to ␦ ␦ check that it works: ͵ d4xd4xЈC͑x,xЈ͒ 4͑y͒ ␦͑x͒ ␦͑xЈ͒ ⌽͑ ͒ ⌽͑ ͒ ⌽͑ ͒ ⌽͑ ͒ ⌽͑ ͒ : f1 ... fn+1 :=: f1 ... fn : fn+1 ␦ n = ͵ d4xd4xЈC͑x,xЈ͒ 43͑y͒␦͑xЈ − y͒, ͑51a͒ ͑ ͒ ⌽͑ ͒ ⌽͑ ͒ ␦͑x͒ − ͚ C fk, fn+1 : f1 ... fk−1 k=1 ϫ⌽͑ ͒ ⌽͑ ͒ ͑ ͒ fk+1 ... fn : 44 =͵ d4x,C͑x,y͒122͑y͒␦͑x − y͒ =122͑y͒, ͑51b͒
͵ d ͑⌽͒:⌽͑f ͒ ...⌽͑f ͒:=0, ͑45͒ if the covariance is normalized to unity. We use this result in C 1 n the term: ␦ ␦ ͵ d4xd4xЈ,C͑x,xЈ͒ 122͑y͒ ͵ ͑⌽͒ ⌽͑ ͒ ⌽͑ ͒ ⌽͑ ͒ ⌽͑ ͒ ␦͑x͒ ␦͑xЈ͒ d C : f1 ... fn :: g1 ... gm :=0 ␦ for n m. ͑46͒ = ͵ d4xd4xЈC͑x,xЈ͒ 24͑y͒␦͑xЈ − y͒, ͑52a͒ ␦͑x͒
69 Am. J. Phys., Vol. 76, No. 1, January 2008 Alexander Wurm and Marcus Berg 69 dimensions; that is, the solution of the two-dimensional =24͵ d4xC͑x,y͒␦͑x − y͒ =24, ͑52b͒ Laplace equation is a logarithm. To regulate divergences when p→ϱ, we introduce a cutoff ⌳ for the momentum, with the same normalization of the covariance. We collect which makes C=ln⌳. If we substitute this result into Eq. these results with the appropriate coefﬁcients from the ex- ͑55b͒, we obtain pansion: 2 :eip:=⌳p /2eip. ͑56͒ 1 1 1 4͑ ͒ 4͑ ͒ 2͑ ͒ ͑ ͒ : y :C = y − ·12 y + 2 ·24, 53a Thus the anomalous scaling dimension, usually denoted by 2 2! 2 ␥,is␥=p2 /2 for the exponential operator. This result is im- portant in conformal ﬁeld theory ͑see for example Ref. 15͒. 4͑ ͒ 2͑ ͒ ͓͑ ͔͒ ͑ ͒ = y −6 y +3=He4 y , 53b Here the p2 /2 comes from the ␣2 /2 in the generating func- ͑ ͒ which completes the example. tion Eq. 54 . How could such a quantum effect be measured? Consider ͑ C. Further reading the two-dimensional Ising model with random bonds Ref. 13,p.719͒. This model is the familiar Ising model, but the Although we hope to have given some ﬂavor of some of coupling between the spins is allowed to ﬂuctuate, that is, the the techniques and ideas of quantum ﬁeld theory coupling becomes a space-dependent Euclidean ﬁeld. The mathematical-physics style, we have given only a few ex- energy of the system is described by ͑the real part of͒ the amples and demonstrated some simple identities. For more exponential operator :eip:. The anomalous dimension in Eq. on the mathematical connection between Wick transforms on ͑56͒ leads to an expression for the speciﬁc heat ͓see Ref. 13, function spaces and Wick-ordering of annihilation and cre- Eq. ͑356͔͒, where the derivation is given using renormaliza- ation operators, we recommend Refs. 7 and 8. For readers tion group methods. The speciﬁc heat is in principle directly more interested in fermions than the scalar ﬁelds we have measurable as a function of the temperature, or more conve- ͑ ͒/ discussed, we recommend Ref. 14 as an introduction to Wick niently, as a function of = T−Tc Tc, a dimensionless de- ordering. viation from the critical temperature. The renormalization group description predicts a double logarithm dependence on D. An application: Speciﬁc heat that could not have been found by simple perturbation theory; it uses as input the result from Eq. ͑56͒. In this section we discuss an example of a physical appli- Admittedly, the telegraphic description in the previous cation of some of the results we have discussed. By using the ͑ ͒ paragraph does not do justice to the full calculation of the connection Eq. 31 between the Wick transform and Her- speciﬁc heat in the two-dimensional Ising model with ran- mite polynomials, we show how standard properties of those dom bonds. Our purpose here is only to show how the Wick polynomials can be exploited to simplify certain calcula- transform reproduces the quantum effect in Eq. ͑56͒, and tions. then to give some ﬂavor of how this effect is measurable. Consider the familiar generating function of Hermite poly- nomials ͓but for the scaled polynomials Eq. ͑16͔͒: IV. CONCLUSION ␣n x␣−␣2/2 ͑ ͒ ͑ ͒ e = ͚ Hen x . 54 We have shown that the scope of normal ordering has n! expanded to settings beyond the original one of ordering This generating function gives a shortcut to calculations in operators and we have discussed an interesting connection to two-dimensional quantum ﬁeld theory, where normal order- Hermite polynomials, which is usually not mentioned in ing is often the only form of renormalization necessary. An courses on quantum ﬁeld theory. Several different deﬁnitions important issue is the scaling dimension of the normal- of Wick ordering of functions have been discussed and their ordered exponential :eip:, where p is a momentum. ͑The equivalence established. real part of this operator can represent the energy of a system From a physics point of view, Wick ordering is part of the where is the quantum ﬁeld.͒ In other words, the question is process of renormalization, which systematically removes in- if we rescale the momentum p→⌳p, equivalent to a rescal- ﬁnities from a theory ͑that is, removes inﬁnite terms from the ing x→⌳−1x in coordinate space, how does the operator perturbation expansion of the functional integral͒. In general, :eip: scale? Because an exponential is dimensionless, we Wick ordering needs to be supplemented by additional rules might guess that the answer is that it does not scale at all, for renormalization ͑see for example, Ref. 2͒ From a math- that is, the scaling dimension is zero. This answer is not so ematics point of view, Wick ordering is helpful to make due to quantum effects induced by the normal ordering. In sense of the polynomials of ﬁelds that are integrated over in terms of the functional integral we can easily calculate the the functional integral. effect of the normal-ordering ͑here, the Wick transform͒: For a deeper understanding and further applications of ϱ these ideas, the interested reader is invited to consult the 1 ip ͑ ͒n n ͑ ͒ cited literature, which is a selection of texts we have found :e :C = ͚ ip : :C, 55a particularly useful. In particular, for the physics of functional n=0 n! integrals we recommend Ref. 7. For a more mathematically ϱ oriented treatment we have found Ref. 8 to be quite useful. 1 1 2 ͑ ͒n n/2 ͑/ͱ ͒ p C ip ͑ ͒ =͚ ip C Hen C = e 2 e , 55b n=0 n! ACKNOWLEDGMENTS where we used the deﬁnition Eq. ͑31͒ and the generating We would like to thank C. DeWitt–Morette, P. Cartier, and function Eq. ͑54͒. The covariance C is a logarithm in two D. Brydges for many helpful discussions.
70 Am. J. Phys., Vol. 76, No. 1, January 2008 Alexander Wurm and Marcus Berg 70 APPENDIX A: WICK-ORDERING OF OPERATORS 1 1 ˆ ͑ † †͒ ͩ † ͪ ͑ ͒ IN QUANTUM FIELD THEORY H = ͚ akak + akak k = ͚ akak + k, A7 2 k k 2 We brieﬂy remind the reader how the need for Wick or- where in the last step we used the commutation relations dering arises in the operator formulation of quantum ﬁeld from Eq. ͑A5c͒. A calculation of the vacuum energy reveals theory. All of this material is standard and can be found in a potential problem: introductory books on quantum ﬁeld theory ͑for example, 1 1 Ref. 2͒, albeit in lengthier and more thorough form. We set ͗ ͉ ˆ ͉ ͘ ͗ ͉ ͚͘ ͚ → ϱ ͑ ͒ ប 0 H 0 = 0 0 k = k , A8 =1 throughout. k 2 k 2 Consider a real scalar ﬁeld ͑t,x͒ of mass m deﬁned at all ͗ ͉ ͘ points of four-dimensional Minkowski spacetime and satis- where we have used the normalization condition 0 0 =1. fying the Klein-Gordon equation: This inﬁnite constant can be removed as described in the text. Propagation amplitudes in quantum ﬁeld theory ͑and 2ץ m2ͪ͑t,x͒ =0. ͑A1͒ + 2ٌ − ͩ t2 hence scattering cross sections and decay constants͒ areץ given in terms of expectation values of time-ordered prod- The differential operator in parenthesis is one instance of ucts of ﬁeld operators. These time-ordered products arise in what we call D in the text. The classical Hamiltonian of this the interaction Hamiltonian of an interacting quantum ﬁeld scalar ﬁeld is: theory. The goal is to calculate propagation amplitudes for these interactions using time-dependent perturbation theory 1 familiar from quantum mechanics. To leading order in the H = ͚ ͓͑͑t,x͒͒2 + ٌ͑͑t,x͒͒2 + m22͑t,x͔͒, ͑A2͒ 2 x coupling constant, these products can be simpliﬁed, and the zero-point constant energy removed by using Wick’s theo- where is the variable canonically conjugate to , rem. -t. We can think of the ﬁrst term as the kinetic energy The way Wick ordering is applied in practice to calculaץ/ץ= and the second as the shear energy. This classical system is tions in quantum ﬁeld theory is through Wick’s theorem, quantized in the canonical quantization scheme by treating which gives a decomposition of time-ordered products of the ﬁeld as an operator, and imposing equal-time commu- ﬁeld operators into sums of normal-ordered products of ﬁeld tation relations: operators ͑again, we refer to Ref. 2͒. Wick’s theorem appears in the functional-integral formulation of the theory in Sec. ͓͑t,x͒,͑t,xЈ͔͒ =0, ͑A3a͒ IIIA3.
͓͑t,x͒,͑t,xЈ͔͒ =0, ͑A3b͒ APPENDIX B: PROPERTIES OF Hen„x… ͓͑t,x͒,͑t,xЈ͔͒ = i␦3͑x − xЈ͒. ͑A3c͒ Here we list a few useful properties of the scaled Hermite polynomials. More properties can be found in Ref. 5. The plane-wave solutions of the Klein–Gordon equation are Deﬁning series: ͑ ͒ known as the ﬁeld modes, uk t,x . Together with their re- ͓ n ͔ *͑ ͒ 2 spective complex conjugates uk t,x they form a complete n! He ͑x͒ = ͚ ͑−1͒m xn−2m, ͑B1͒ orthonormal basis. Hence, the ﬁeld can be expanded as n m͑ ͒ m=0 m !2 n −2m ! ͑ ͒ ͓ ͑ ͒ † *͑ ͔͒ ͑ ͒ t,x = ͚ akuk t,x + akuk t,x . A4 where ͓n/2͔ is the integer part of n/2. k Orthogonality: The equal time commutation relations for and are then ϱ ͵ −x2/2 ͑ ͒ ͑ ͒ ␦ ͱ ͑ ͒ equivalent to dxe Hen x Hem x = mn 2 n !. B2 −ϱ ͓a ,a ͔ =0, ͑A5a͒ k kЈ Generating function: ϱ † ␣n ͓ † ͔ ͑ ͒ 1 ak,a =0, A5b ͑ ␣ ␣2͒ ͑ ͒ ͑ ͒ kЈ exp x − 2 = ͚ Hen x . B3 n=0 n! ͓ † ͔ ␦ ͑ ͒ ak,akЈ = kkЈ. A5c Recursion relation: ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ These operators are deﬁned on a Fock space, which is a Hen+1 x = xHen x − nHen−1 x . B4 Hilbert space made of n-particle states ͑n=0,1,...͒. The The ﬁrst ﬁve polynomials are normalized basis ket vectors, denoted by ͉͘, can be con- He ͑x͒ =1, ͑B5a͒ structed starting from the vector ͉0͘. The vacuum state ͉0͘ 0 has the property that it is annihilated by all the ak operators: ͑ ͒ ͑ ͒ He1 x = x, B5b ͉ ͘ ∀ ͑ ͒ ak 0 =0, k. A6 ͑ ͒ 2 ͑ ͒ He2 x = x −1, B5c ͱ͉ ͉2 2 In terms of the frequency k =c k +m , the Hamiltonian ͑ ͒ ͑ ͒ 3 ͑ ͒ operator obtained from Eq. A2 is He3 x = x −3x, B5d
71 Am. J. Phys., Vol. 76, No. 1, January 2008 Alexander Wurm and Marcus Berg 71 He ͑x͒ = x4 −6x2 +3. ͑B5e͒ Verlag, New York, 1981͒. 4 8 Svante Janson, Gaussian Hilbert Spaces ͑Cambridge University Press, Cambridge, 1997͒. ͒ 9 ͑ a Electronic address: [email protected] To avoid dimension-dependent numerical terms powers of , powers of ͒ ͑ ͒ 1 G. C. Wick, “The evaluation of the collision matrix,” Phys. Rev. 80, 2 in the deﬁnition 24 of the Gaussian measure we can, alternatively, ͐ d /͑ ͒1/2 −Q͑x͒ −2i͗xЈ,x͘ −W͑xЈ͒ 268–272 ͑1950͒. deﬁne it by Rdd x det D e e =e . This deﬁnition 2 Michael Peskin and Daniel V. Schroeder, An Introduction to Quantum can be convenient in the simplest Fourier transforms for those who forget Field Theory ͑Addison-Wesley, Reading, MA, 1995͒. See in particular where the 2 goes: ˆf͑p͒=͐dxe−2ipxf͑x͒ yields an inverse f͑x͒ the chapter on functional integrals in quantum ﬁeld theory. =͐dpe2ipxˆf͑p͒, without the prefactor. 3 ͑ 10 ͑͒ ͑ Lewis H. Ryder, Quantum Field Theory Cambridge University Press, Barry Simon, The P 2 Euclidean (Quantum) Field Theory Princeton Cambridge, 1985͒. University Press, Princeton, NJ, 1974͒. 4 Jun J. Sakurai, Modern Quantum Mechanics, rev. ed. ͑Addison-Wesley, 11 Similar to the division of power series, the multiplication of a power ͒ ͚͑ϱ k͚͒͑ϱ k͒ ͚ϱ n ͚n Reading, MA, 1994 . series is k=0akx k=0bkx = n=0dnx , where dn = m=0ambn−m. 5 Milton Abramovitz and Irene A. Stegun, Handbook of Mathematical 12 Joseph Polchinski, String Theory: An Introduction to the Bosonic String, Functions ͑Dover, New York, 1965͒. Vol. 1 ͑Cambridge University Press, Cambridge, 1998͒. 6 In particular, contributions to the S-matrix that have an essential singu- 13 Claude Itzykson and Jean-Michel Drouffe, Statistical Field Theory, Vols. larity at zero coupling constant cannot be found by standard perturbative 1 and 2 ͑Cambridge University Press, Cambridge, 1989͒. expansion around zero coupling. Yet these “nonperturbative” contribu- 14 Manfred Salmhofer, Renormalization: An Introduction ͑Springer-Verlag, tions can be studied using functional integrals. Rates for decays that Berlin, 1999͒. would be strictly forbidden without these effects can be calculated; see 15 Edward Witten, “Perturbative quantum ﬁeld theory,” in Quantum Fields for example, Ref. 3, Sec. 10.5. and Strings: A Course for Mathematicians, edited by P. Deligne et al. 7 James Glimme and Arthur Jaffe, Quantum Physics, 2nd ed. ͑Springer- ͑American Mathematical Society, Providence, RI, 1999͒, Vol. 1, p. 451.
Inertia Demonstration. We still do this demonstration today. A stiff card is placed on a pillar and a marble is balanced atop the card. A quick blow with the hand or a snap with the ﬁngertip on the edge of the card knocks it out of the way, and the marble drops into a small depression in the top of the tower. For reliable demonstrations, the digit is replaced by a strip of spring steel. This example is at Grinnell College in Iowa. Suggestion: make a half dozen of these ͑they can be made with blocks of wood and a section of an old hacksaw blade͒ and put them in the introductory lab for students to play with before lab starts. ͑Photograph and Notes by Thomas B. Greenslade, Jr., Kenyon College͒
72 Am. J. Phys., Vol. 76, No. 1, January 2008 Alexander Wurm and Marcus Berg 72