Simplified Dirac Delta

7/27/2019 Simplified Dirac Delta

1/18

Simplified production of

DIRAC DELTA FUNCTION IDENTITIES

Nicholas Wheeler, Reed College Physics Department

November 1997

Introduction. To describe the smooth distribution of (say) a unit mass on thex-axis, we introduce distribution function (x) with the understanding that

(x) dx mass element dm in the neighborhood dx of the point x(x) dx = 1

To describe a mass distribution localized to the vicinity of x = a we might, forexample, write

(x a; ) =

12 if a < x < a + , and 0 otherwise; else

12 exp

12 (x a)2; else1x

sin(x/); else . . .

In each of those cases we have

(x a; ) dx = 1 for all > 0, and in eachcase it makes formal sense to suppose that

lim0

(x a; ) describes a unit point mass situated at x = a

Dirac clearly had precisely such ideas in mind when, in 15 of his QuantumMechanics,1 he introduced the point-distribution (x a). He was well aware

1 I work from his Revised 4th Edition (), but the text is unchanged fromthe 3rd Edition (). Diracs first use of the -function occurred in a paperpublished in , where (x y) was intended to serve as a continuous analogof the Kronecker delta mn, and thus to permit unified discussion of discreteand continuous spectra.


2/18

2 Simplified Dirac identities

that the delta functionwhich he presumes to satisfy the conditions

+

(x a) dx = 0(x a) = 0 for x = a

is not a function. . . according to the usual mathematical definition; it is,in his terminology, an improper function, a notational device intended toby-pass distracting circumlocutions, the use of which must be confined tocertain simple types of expression for which it is obvious that no inconsistencycan arise.

Diracs cautionary remarks (and the efficient simplicity of his idea)notwithstanding, some mathematically well-bred people did from the outsettake strong exception to the -function. In the vanguard of this group wasJohn von Neumann, who dismissed the -function as a fiction, and wrotehis monumental Mathematische Grundlagen der Quantenmechanik2 largely todemonstrate that quantum mechanics can (with sufficient effort!) be formulatedin such a way as to make no reference to such a fiction.

The situation changed, however, in , when Laurent Schwartz publishedthe first volume of his demanding multi-volume Theorie des distributions.Schwartz accomplishment was to show that -functions are (not functions,either proper or improper, but) mathematical objects of a fundamentally newtypedistributions, that live always in the shade of an implied integral sign.This was comforting news for the physicists who had by then been contentedlyusing -functions for thirty years. But it was news without major consequence,for Schwartz work remained inaccessible to all but the most determined ofmathematical physicists.

Thus there came into being a tradition of simplification and popularization.In Schwartz gave a series of lectures at the Seminar of the CanadianMathematical Congress (held in Vancouver, B.C.), which gave rise in toa pamphlet3 that circulated widely, and brought at least the essential elementsof the theory of distributions into such clear focus as to serve the simple needsof non-specialists. In the British applied mathematician G. Templebuilding upon remarks published a few years earlier by Mikusnski4publishedwhat he called a less cumbersome vulgarization of Schwartz theory, whichhe hoped might better serve the practical needs of engineers and physicists.Temples lucid paper inspired M. J. Lighthill to write the monograph fromwhich many of the more recent introductions to the theory of distributions

2 The German edition appeared in . I work from the English translationof . Remarks concerning the -function can be found in 3 of Chapter I.

3 I. Halperin, Introduction to the Theory of Distributions.4 J. G. Mikusnski, Sur la methode de generalization de Laurent Schwartz

et sur la convergence faible, Fundamenta Mathematicae 35, 235 (1948).


3/18

Introduction 3

descend. Lighthills slender volume5by intention a text for undergraduatesbears this dedication

to

paul diracwho saw that it must be true

laurent schwartz

who proved it,and

george temple

who showed how simple it could be made

and has about itas its title promisesa distinctly Fourier analytic flavor.Nor is this fact particularly surprising; the Fourier integral theorem

f(x) = 12

+

eikx+

e+ikyf(y) dy

dk for nice functions f()

can by reorganization be read as an assertion that

12

+

eik(yx) dk = (y x)

The history of the -function can in this sense be traced back to the earlys. Fourier, of course, was concerned with the theory of heat conduction,but by the -function had intruded for a second time into a physical theory;George Green noticed that the solution of the Poisson equation 2(x) = (x),considered to describe the electrostatic potential generated by a given chargedistribution (x), can be obtained by superposition of the potentials generatedby a population of point charges; i.e., that the general problem can be reduced

to the special problem 2(x; y) = (x y)where now the -function is being used to describe a unit point chargepositioned at the point y. Thus came into being the theory of Greensfunctions, whichwith important input by Kirchhoff (physical optics, in thes) and Heaviside (transmission lines, in the s)became, as it remains,one of the principal consumers of applied distribution theory.

I have sketched this history6 in order to make clear that what I proposeto do in these pages stands quite apart, both in spirit and by intent, from thetrend of recent developments, and is fashioned from much ruder fabric. Myobjective is to promote a point of viewa computational techniquethat came

5 Introduction to Fourier Analysis & Generalized Functions ().6 Of which Jesper Lutzen, in his absorbing The Prehistory of the Theory of

Distributions(), provides a wonderfully detailed account. In his ConcludingRemarks Lutzen provides a nicely balanced account of the relative contributionsof Schwartz and of S. L. Sobelev (in the early s).


4/18


accidentally to my attention in the course of work having to do with the one-dimensional theory of waves.7 I proceed very informally, and will be concernednot at all with precise characterization of the conditions under which the things

I have to say may be true; this fact in itself serves to separate me from recenttradition in the field.

Regarding my specific objectives. . . Dirac remarks that There are anumber of elementary equations which one can write down about functions.These equations are essentially rules of manipulation for algebraic workinvolving functions. The meaning of any of these equations is that its twosides give equivalent results [when used] as factors in an integrand. Examplesof such equations are

(x) = (x)x(x) = 0

(ax) = a1(x) : a > 0 (1.1)

(x2 a2) = 12 a1(x a) + (x + a) : a > 0 (1.2)(a x) dx (x b) = (a b)

f(x)(x a) = f(a)(x a)

On the evidence of this list (which attains the length quoted only in the3rd edition) Lutzen concludes that Dirac was a skillful manipulator of the-function, and goes on to observe that some of the above theorems, especially(1.2), are not even obvious in distribution theory, since the changes of variablesare hard to perform. . . 8 The formal identities in Diracs list are of severaldistinct types; he supplies an outline of the supporting argument in all casesbut one: concerning (1) he remarks only that they may be verified by similarelementary arguments. But the elementary argument that makes such easy

work of (1.1)9 acquires a fussy aspect when applied to expressions of the form

g(x)

typified by the left side of (1.2). My initial objective will be todemonstrate that certain kinds of -identities (including particularly those oftype (1)) become trivialities when thought of as corollaries of their-analogs. Byextension of the method, I will then derive relationships among the derivativeproperties of () which are important to the theory of Greens functions.

7 See R. Platais, An investigation of the acoustics of the flute (Reed Collegephysics thesis, ).

8 See Chapter 4, 29 in the monograph previously cited.9 By change of variables we have

f(x)(ax) dx = f(y/a)(y) 1|a|dy = 1|a|f(0)=

f(y) 1|a|(y) dy

which assumes only that the Jacobian |a| = 0.


5/18

Heaviside step function 5

1. Properties and applications of the Heaviside step function.The step function()introduced by Heaviside to model the action of a simple switchcan bedefined

(x) = 0 for x < 012 at x = 0

1 for x > 0(2)

where the central 12 is a (usually inconsequential) formal detail, equivalent tothe stipulation that

(x) 2(x) 1 =

1 for x < 00 at x = 0

+1 for x > 0(3)

be odd (i.e., that () vanish at the origin). As Dirac himself (and before himHeaviside) have remarked, the step function (with which Dirac surely becameacquainted as a student of electrical engineering) and the -function stand in aclose relationship supplied by the calculus:

(x a) =x

(y a) dy (4.1)

ddx

(x a) = (x a) (4.2)The central 12 is, in this light, equivalent to the stipulation that (x) be(formally) an even function of x. For the same reason that (x) becomesmeaningful only in the shade of an integral sign, so also does (x), at leastas it is used in intended applications; the construction

(x) lim0

(x; ) entails (x) = lim0

(x; )

x

(y; ) dy

and causes (x; ) to become literally differentiable at the origin, except in the

limit.

That (x) and (x) are complementary constructs can be seen in yetanother way. The identity

f(x) =

+

(x y)f(y) dy (5)

provides what might be called the picket fence representation of f(x). But

=

+

ddy

(x y)f(y) dy=

+

(x y)f(y) dy + boundary term (6)

which (under conditions that cause the boundary term to vanish) provides theless frequently encountered stacked slab representation off(x). In the formerit is f() itself that serves to regulate the heights of successive pickets, whilein the latter it is not f but its derivative f

() that regulates the thicknessesof successive slabs. For graphical representations of (5) and (6) see Figure 1.


6/18


Figure 1: The picket fence representation (5) off(x), comparedwith the stacked slab representation (6).

Partial integration (subject always to the presumption that boundary termsvanish), which we used to obtain (6), is standardly used also to assign meaning

to the successive derivatives of the -function; one writesf(y)(y x) dy =

f(y)(y x) dy = f(x)

f(y)(y x) dy =

f(y)(y x) dy

= ()2

f(y)(y x) dy = ()2f(x)...

f(y)(n)(y x) dy = ()nf(n)(x)

Formally, one has (see Figure 2)

(y x) = lim0

y (x ) y (x + )2

(y x) = lim0

y (x 2) 2y x) + y (x + 2)(2)2

...

(7)

which when introduced as factors into the integrand of

f(y) dy serve simplyto reconstruct the definitions of f(x), f(x), etc.

In anticipation of future needs, I note that the preceding equations canbe formulated as relations among -functions. Taking (4.2) as our point ofdeparture, we have

(y x) = lim0

y (x ) y (x + )2

= limit of a sequence of ever-narrower/taller box functions


7/18

Simplified derivation of delta function identities 7

xy

x

Figure 2: The figures on the left derive from (7), and show representations of ascending derivatives of (y x). The figureson the right derive from (8), and provide representations of thesame material.

giving (see again the preceding figure)

(y x) = lim0

(y[x2])2(yx)+(y[x+2])(2)2

(y x) = lim0

(y[x3])3(y[x])+3(y[x+])(y[x+3])(2)3

...

(8)

2. Simplified derivation of delta function identities. Let (x; ) refer to some(any nice) parameterized sequence of functions convergent to (x), and let a

be a positive constant. While (ax; ) and (x; ) are distinct functions ofx, they clearly become identical in the limit 0, and so also therefore dotheir derivatives (of all orders). So we have a (ax) = (x), which by (4.2)reproduces (1.1):

(ax) = a1(x) : a > 0 (9.1)


8/18


If, on the other hand, a < 0 then (ax) = 1 (x) gives a (ax) = (x)whence

(ax) = a1(x) : a < 0 (9.2)The equation

(ax) = 1|a| (x) : a = 0 (10.1)provides a unified formulation of (9.1) and (9.2). A second differentiation gives

a2 (ax) = (x) according as a 0

(ax) = 1a2

(x) (10.2)

Preceding remarks illustrate the sense in which -identities become trivialitieswhen thought of as corollaries of their -analogs. By way of more interestingillustration of the same point. . .

The function g(x) x2 a2 describes an up-turned parabola that crossesthe x-axis at x = a. Evidently

(x2 a2) =

1 for x < a0 for a < x < +a1 for x > +a

= 1 (x + a) (x a)by differentiation entails

2x (x2 a2) = (x + a) + (x a)

(x2

a2

) = (2a)1

(x + a) + (+2a)1

(x a)

= 12|a|

(x a) + (x + a)

(11)

which provides a generalized formulation if (1.2). By this mode of argument itbecomes transparently clear how the a that enters into the prefactor comes toacquire its (otherwise perplexing) absolute value braces.

Suppose, more generally, that

g(x) = g0(x x1)(x x2) (x xn)

with x1 < x2 t (forward lightcone) it is a triangular

excavation of similar design.

Greens method leads one to write

F

(t, x) =

(x y, t u)F(u, y) dydu

and to require of (x y, t u) that

2t 2x

(x y, t u) = (y x)(u t) (17)

and (y x, 0) = 0. Such properties are possessed by (in particular) this closerelative of 0

(x y, t u) = (t u) 0(x y, t u) (18)

= 1

2(y x) + (t u) (y x) (t u) : t u > 00 : t u < 0=

+ 12 inside the lightcone extending backward from {t, x}0 inside the lightcone extending forward from {t, x}0 outside the lightcone


12/18


which is known as the retarded propagator. The properties in questionare possessed also by the similarly-defined advanced propagator, and aretherefore shared also by all functions of the form11

(, ) = kret(, ) + (1 k)adv(, )

We encounter at this point the need for a principle of choice, but it woulddistract me from my primary objective to pursue the matter.

Concerning that primary objective: I entered into the preceding reviewof what is after all standard material partly to demonstrate that the Fourieranalytic methods standardly encountered12 are, in fact, inessential (the subjectis susceptible to development by elementary means). . . but mainly to place

Figure 5: Representation of the truncation process (18), bymeans of which 0 gives rise to .

myself in position to pose this question: What is the detailed mechanism bymeans of which truncation causes a solution 0 of the homogeneous waveequation to become a solution of the inhomogeneous equation:

0 = 0 truncation of 0

=

Elementary insight into the geometrical essence of the answer is provided bythe following sequence of figures:

11 See F. Rohrlich, Classical Charged Particles (), p. 79.12 See, for example, relativistic classical fields (), pp. 163190 and

analytical methods of physics (), pp. 291297.


13/18

Elementary geometrical theory of Greens functions 13

-1

0

+1

-1

t

+1

x

Figure 6: The numbers, when multiplied by 12 (2)2, referin the

sense familiar from (7) and (8); see also the lower right detail inFigure 2to the discrete approximation

+2t (x y, t u)

+ (xy,[t+2]u) 2(xy,[t]u)+(xy,[t2]u)(2)2

In this and subsequent figures (x y, t u) is considered to be afunction of{u, y}, into which {t, x} enter as parameters.


14/18


+1

-1

0

+1

-1

t

x

+2

Figure 7: The numbers, when multiplied by 12 (2)2, refer to the

discrete approximation

2x(x y, t u)

([x+2]y,tu) 2([x]y,tu)+([x2]y,tu)(2)2


15/18

Elementary geometrical theory of Greens functions 15

0

+1

t

x

+1

000

0

Figure 8: Superposition of the preceding figures. The numbers,when multiplied by 12 (2)

2, supply the discrete approximation2t 2x

(x y, t u) 12 (2)2 Box(t, x; )

where

Box(t, x; )

1 interior to shaded box centered at (t, x)0 elsewhere

The box has diagonal measure 4, and its area is given therefore by(4/

2)2 = 82. It is the top of a little prism of height 12 (2)

2, thevolume of which is given by

volume = 1 (all values of)

evidently

lim0

12 (2)

2 Box(t, x; ) = (u t)(y x)

which yields the result we sought to establish.


16/18


4. Analytical derivation of the same result. The discussion in the precedingsection culminated in a geometrical argument designed to illuminate how itcomes about that

0 = 0 but truncation engenders =

For comparative purposes I now outline the analytical demonstration of thatsame fact; the details are not without interest, but the argument as a wholeseems to me to lack the ah-ha! quality of its geometrical counterpart.

By (18) we have

(x y, t u) = 12 (t u)(&c.)&c. (t u)2 (x y)2

Therefore

t =12 (t

u)(&c.) + (t

u)(t

u)(&c.)

tt =12

(t u)(&c.) + 2(t u)(t u)(&c.)+ (t u)(&c.) + (t u)2(t u)2(&c.)

x = (t u)(x y)(&c.)xx = (t u)(&c.) + (t u)2(x y)2(&c.)

giving2t 2x

= 12

(t u)(&c.) + 2(t u)(t u)(&c.)+ 2(t u)(&c.) + (&c.)(&c.)

0where the term that vanishes does so because x(x) = 0 = (x) + x(x) = 0.We note in passing that had we omitted the -factor from the definition of ;i.e., if we were evaluating 0 instead of , we would at this point haveachieved 0 = 0.

Our assignment now is to establish that

12

(t u)(&c.) + 2(t u)(t u)(&c.) = (t u)(x y)which, if we write t u and x y, can be notated

12

()(2 2) + 2() (2 2)

= ()()

But = 12

()(2 2) +() (2 2)

I will argue that= 0 (19)


17/18

Dimensional generalization 17

Assuming, for the moment, the truth of that claim, we want to show that

() (2 2)

= ()() ()

But this is in fact immediate; looking to (11) and taking advantage ofsimplifications made available by the presence of the ()-factor, we have

= () 12

( ) + (+ )= () 12

() + ()

= () ()

Returning now to the demonstration of (19); if F() is nice function, then(integrating by parts) we have

()(2 2)F() d = ()(2 2)F() d

=

()

1 ( + ) ( )F() d by (12)=

()F() d +

+

()F() d

= 0 all > 0, also in the limit 0 because F() is nice

The argument just concluded is notable for its delicacy 13 and its overallimprovisatory, ad hoc quality. It is, in my view, not only less illuminating butalso less convincing than the geometrical argument summarized in Figures 68.It points up the need for a systematic calculus of distributions, and lendsforce to Diracs observation that we should (in the absence of such a calculus)avail ourselves of-methods only when it is obvious that no inconsistency canarise. This, however, is more easily said than done.

5. Relaxation of the assumption that spacetime is 2- dimensional. In 3 wewere able to obtain Greens functions of the homogeneous/inhomogeneous waveequations

2t 2x

0 = 02t 2x

=

by putting dAlembert and Heaviside/Dirac in a pot and stirring gently, thenotable fact being that no Fourier was called for by the recipe. dAlembert madecritical use, however, of circumstances which are special to the 2-dimensional

13 We would, for example, have gone off-track if we had made too-casualuse of Diracs identity x(x) = 0; this makes good sense in the context Diracintended (

x(x) f(x) dx = 0), but can lead to error when (as at ()) x(x)

appears as a factor in a more complex expression.


18/18

Simplified Dirac Delta

Documents

Transcript of Simplified Dirac Delta