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DIRAC DELTA FUNCTION IDENTITIES
Nicholas Wheeler, Reed College Physics Department November 1997
Introduction. To describe the smooth distribution of (say) a unit mass on the x-axis, we introduce distribution function μ(x) with the understanding that
μ(x) dx ≡ mass element dm in the neighborhood dx of the point x ∫ μ(x) dx = 1
To describe a mass distribution localized to the vicinity of x = a we might, for example, write
μ(x − a; ) =
1 2 if^ a^ −^ < x < a^ +^ , and 0 otherwise; else
√^1 2 π exp^
− (^21) (x − a)^2
; else
1 πx sin(x/);^ else^...
In each of those cases we have
μ(x − a; ) dx = 1 for all > 0, and in each case it makes formal sense to suppose that
lim ↓ 0 μ(x − a; ) describes a unit point mass situated at x = a
Dirac clearly had precisely such ideas in mind when, in §15 of his Quantum Mechanics,^1 he introduced the point-distribution δ(x − a). He was well aware
(^1) I work from his Revised 4th (^) Edition (), but the text is unchanged from
the 3rd^ Edition (). Dirac’s first use of the δ-function occurred in a paper published in , where δ(x − y) was intended to serve as a continuous analog of the Kronecker delta δmn, and thus to permit unified discussion of discrete and continuous spectra.
2 Simplified Dirac identities
that the “delta function”—which he presumes to satisfy the conditions
∫ (^) +∞
−∞
δ(x − a) dx = 0
δ(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 to by-pass distracting circumlocutions, the use of which “must be confined to certain simple types of expression for which it is obvious that no inconsistency can arise.”
Dirac’s cautionary remarks (and the efficient simplicity of his idea) notwithstanding, some mathematically well-bred people did from the outset take strong exception to the δ-function. In the vanguard of this group was John von Neumann, who dismissed the δ-function as a “fiction,” and wrote his monumental Mathematische Grundlagen der Quantenmechanik 2 largely to demonstrate that quantum mechanics can (with sufficient effort!) be formulated in such a way as to make no reference to such a fiction.
The situation changed, however, in , when Laurent Schwartz published the first volume of his demanding multi-volume Th´eorie des distributions. Schwartz’ accomplishment was to show that δ-functions are (not “functions,” either proper or “improper,” but) mathematical objects of a fundamentally new type—“distributions,” that live always in the shade of an implied integral sign. This was comforting news for the physicists who had by then been contentedly using δ-functions for thirty years. But it was news without major consequence, for Schwartz’ work remained inaccessible to all but the most determined of mathematical physicists.
Thus there came into being a tradition of simplification and popularization. In Schwartz gave a series of lectures at the Seminar of the Canadian Mathematical Congress (held in Vancouver, B.C.), which gave rise in to a pamphlet^3 that circulated widely, and brought at least the essential elements of the theory of distributions into such clear focus as to serve the simple needs of non-specialists. In the British applied mathematician G. Temple— building upon remarks published a few years earlier by Mikus´ınski^4 —published what he called a “less cumbersome vulgarization” of Schwartz’ theory, which he hoped might better serve the practical needs of engineers and physicists. Temple’s lucid paper inspired M. J. Lighthill to write the monograph from which many of the more recent “introductions to the theory of distributions”
(^2) The German edition appeared in . I work from the English translation
of . Remarks concerning the δ-function can be found in §3 of Chapter I. (^3) I. Halperin, Introduction to the Theory of Distributions. (^4) J. G. Mikus´ınski, “Sur la m´ethode de g´en´eralization de Laurent Schwartz
et sur la convergence faible,” Fundamenta Mathematicae 35 , 235 (1948).
4 Simplified Dirac identities
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 concerned not 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 recent tradition in the field.
Regarding my specific objectives... Dirac remarks that “There are a number of elementary equations which one can write down about δ functions. These equations are essentially rules of manipulation for algebraic work involving δ functions. The meaning of any of these equations is that its two sides give equivalent results [when used] as factors in an integrand. Examples of such equations are
δ(−x) = δ(x) xδ(x) = 0 δ(ax) = a−^1 δ(x) : a > 0 (1.1) δ(x^2 − a^2 ) = 12 a−^1
δ(x − a) + δ(x + a)
: a > 0 (1.2) ∫ δ(a − x) dx δ(x − b) = δ(a − b) f (x)δ(x − a) = f (a)δ(x − a) ”
On the evidence of this list (which attains the length quoted only in the 3 rd^ edition) L¨utzen concludes that “Dirac was a skillful manipulator of the δ-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 variables are hard to perform... ”^8 The formal identities in Dirac’s list are of several distinct types; he supplies an outline of the supporting argument in all cases but one: concerning (1) he remarks only that they “may be verified by similar elementary 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 δ
g(x)
typified by the left side of (1.2). My initial objective will be to demonstrate that certain kinds of δ-identities (including particularly those of type (1)) become trivialities when thought of as corollaries of their θ-analogs. By extension of the method, I will then derive relationships among the derivative properties of δ(•) which are important to the theory of Green’s functions.
(^7) See R. Platais, “An investigation of the acoustics of the flute” (Reed College
physics thesis, ). (^8) See Chapter 4, §29 in the monograph previously cited. (^9) By change of variables we have
∫ f (x)δ(ax) dx =
f (y/a)δ(y) (^) |^1 a| dy = (^) |^1 a| f (0)
f (y) (^) |^1 a| δ(y) dy
which assumes only that the Jacobian |a| = 0.
Heaviside step function 5
1. Properties and applications of the Heaviside step function .The step function θ(•)—introduced by Heaviside to model the action of a simple switch—can be defined
θ(x) =
0 for x < 0 1 2 at^ x^ = 0 1 for x > 0
where the central 12 is a (usually inconsequential) formal detail, equivalent to the stipulation that
ε(x) ≡ 2 θ(x) − 1 =
− 1 for x < 0 0 at x = 0 +1 for x > 0
be odd (i.e., that ε(•) vanish at the origin). As Dirac himself (and before him Heaviside) have remarked, the step function (with which Dirac surely became acquainted as a student of electrical engineering) and the δ-function stand in a close relationship supplied by the calculus:
θ(x − a) =
∫ (^) x
−∞
δ(y − a) dy (4.1)
d dx θ(x^ −^ a) =^ δ(x^ −^ a)^ (4.2)
The central 12 is, in this light, equivalent to the stipulation that δ(x) be (formally) an even function of x. For the same reason that δ(x) becomes meaningful only “in the shade of an integral sign,” so also does θ(x), at least as it is used in intended applications; the construction
δ(x) ≡ lim ↓ 0
δ(x; ) entails θ(x) = lim ↓ 0
θ(x; ) ≡
∫ (^) x
−∞
δ(y; ) dy
and causes θ(x; ) to become literally differentiable at the origin, except in the limit.
That δ(x) and θ(x) are complementary constructs can be seen in yet another way. The identity
f (x) =
−∞
δ(x − y)f (y) dy (5)
provides what might be called the “picket fence representation” of f (x). But
−∞
− ddy θ(x − y)
f (y) dy
−∞
θ(x − y)f ′ (y) dy +boundary term (6)
which (under conditions that cause the boundary term to vanish) provides the less frequently encountered “stacked slab representation” of f (x). In the former it is f (•) itself that serves to regulate the “heights of successive pickets, while in the latter it is not f but its derivative f ′ (•) that regulates the “thicknesses of successive slabs.” For graphical representations of (5) and (6) see Figure 1.
Simplified derivation of delta function identities 7
x
y
x
Figure 2 : The figures on the left derive from (7),and show δ representations of ascending derivatives of δ(y − x). The figures on the right derive from (8),and provide θ representations of the same material.
giving (see again the preceding figure)
δ′(y − x) = lim ↓ 0
θ(y−[x− 2 ])− 2 θ(y−x)+δ(y−[x+2]) (2)^2
δ′′(y − x) = lim ↓ 0
θ(y−[x− 3 ])− 3 θ(y−[x−])+3δ(y−[x+])−θ(y−[x+3]) (2)^3 .. .
2. Simplified derivation of delta function identities. Let θ(x; ) refer to some (any nice) parameterized sequence of functions convergent to θ(x), and let a be a positive constant. While θ(ax; ) and θ(x; ) are distinct functions of x, they clearly become identical in the limit ↓ 0, and so also therefore do their derivatives (of all orders). So we have aθ′(ax) = θ′(x), which by (4.2) reproduces (1.1): δ(ax) = a−^1 δ(x) : a > 0 (9.1)
8 Simplified Dirac identities
If, on the other hand, a < 0 then θ(ax) = 1 − θ(x) gives aθ′(ax) = −θ′(x) whence δ(ax) = −a−^1 δ(x) : a < 0 (9.2)
The equation δ(ax) = (^) |^1 a| δ(x) : a = 0 (10.1)
provides a unified formulation of (9.1) and (9.2). A second differentiation gives
a^2 θ′′(ax) = ±θ′′(x) according as a ≷ 0 ⇓ δ′(ax) = ± (^) a^12 δ′(x) (10.2)
Preceding remarks illustrate the sense in which “δ-identities become trivialities when thought of as corollaries of their θ-analogs.” By way of more interesting illustration of the same point...
The function g(x) ≡ x^2 − a^2 describes an up-turned parabola that crosses the x-axis at x = ±a. Evidently
θ(x^2 − a^2 ) =
1 for x < −a 0 for −a < x < +a 1 for x > +a = 1 −
θ(x + a) − θ(x − a)
by differentiation entails
2 xδ(x^2 − a^2 ) = −δ(x + a) + δ(x − a) ↓ δ(x^2 − a^2 ) = −(− 2 a)−^1 δ(x + a) +(+ 2 a)−^1 δ(x − a)
= (^21) |a|
δ(x − a) + δ(x + a)
which provides a generalized formulation if (1.2). By this mode of argument it becomes transparently clear how the a that enters into the prefactor comes to acquire its (otherwise perplexing) absolute value braces.
Suppose, more generally, that
g(x) = g 0 (x − x 1 )(x − x 2 ) · · · (x − xn)
with x 1 < x 2 < · · · < xn. We then (see Figure 3) have
θ(g(x)) =
θ(x − x 1 ) − θ(x − x 2 ) + · · · − (−)nθ(x − xn)
1 −
θ(x − x 1 ) − θ(x − x 2 ) + · · · − (−)nθ(x − xn)
10 Simplified Dirac identities
But two centuries were to elapse before it came to be widely appreciated that (14) can usefully be notated
ϕ(t , x) =
−∞
∇ 0 (x − y, t)ϕ(0, y) + ∆^0 (x − y, t)ϕt(0, y)
dy (15)
with
∆^0 (x − y, t) ≡ (^12)
θ
y − (x − t)
− θ
y − (x + t)
∇ 0 (x − y, t) ≡ (^12)
δ
y − (x − t)
y − (x + t)
= (^) ∂t∂ ∆^0 (x − y, t)
It is important to notice that, while we had prediction (i.e., evolution from prescribed initial data) in mind when we devised (15), the equation also works retrodictively; (15) is invariant under t −→ −t. Under time-reversal all ∂t’s change sign, but so also (manifestly) does ∆^0 (x− y, t), and when we return with this information to (15) we see that the two effects precisely cancel. Evidently ∆^0 (x − y, t − u)—thought of as a function of {u, y} that depends parametrically on {t , x}—has the properties
∆^0 (x − y, t − u) =
- 12 inside the lightcone extending backward from {t , x} − 12 inside the lightcone extending forward from {t , x} 0 outside the lightcone
= (^12)
θ
(y − x) + (t − u)
− θ
(y − x) − (t − u)
= 12 ε(t − u) · θ
(t − u)^2 − (x − y)^2
illustrated in Figure 4. In order for the right side of (15) to do its assigned work it must be the case that { ∂^2 t − ∂ x^2
∆^0 (x − y, t − u) = 0
and ∆^0 (x − y, 0) = 0 ∆^0 t (x − y, 0) = δ(y − x) ∆^0 tt(x − y, 0) = 0
—all of which are, in fact, quick consequences of (16).
The forced wave equation reads
ϕ = F with F (t , x) prescribed
The general solution can be developed
ϕ = ϕ 0 + ϕF
where ϕ 0 evolves according to ϕ 0 = 0 from prescribed initial data, while ϕF evolves according to ϕF = F but is assumed to vanish on the initial timeslice.
Elementary geometrical theory of Green’s functions 11
t
x
u
y
_
Figure 4: Representation of the Green’s function ∆^0 (x−y, t−u) of the homogeneous wave equation ϕ = 0. For u < t the function has the form of a triangular plateau (backward lightcone) with a flat top at elevation 12 ,while for u > t (forward lightcone) it is a triangular excavation of similar design.
Green’s method leads one to write
ϕF (t , x) =
∆(x − y, t − u)F (u, y) dydu
and to require of ∆(x − y, t − u) that
{ ∂^2 t − ∂ x^2
∆(x − y, t − u) = δ(y − x)δ(u − t) (17)
and ∆(y − x, 0) = 0. Such properties are possessed by (in particular) this close relative of ∆^0
∆(x − y, t − u) = θ(t − u) · ∆^0 (x − y, t − u) (18)
2
[
θ
(y − x) + (t − u)
− θ
(y − x) − (t − u)
)]
: t − u > 0 0 : 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
Elementary geometrical theory of Green’s functions 13
0
t
x
Figure 6: The numbers,when multiplied by 12 (2)−^2 ,refer—in the sense familiar from (7) and (8); see also the lower right detail in Figure 2—to the discrete approximation
+∂ t^2 ∆(x − y, t − u)
≈ +
∆(x−y,[t+2]−u)− 2∆(x−y,[t]−u)+∆(x−y,[t− 2 ]−u) (2)^2
In this and subsequent figures ∆(x − y, t − u) is considered to be a function of {u, y},into which {t , x} enter as parameters.
14 Simplified Dirac identities
0
t
x
Figure 7: The numbers,when multiplied by 12 (2)−^2 ,refer to the discrete approximation
−∂ x^2 ∆(x − y, t − u)
≈ −
∆([x+2]−y,t−u)− 2∆([x]−y,t−u)+∆([x− 2 ]−y,t−u) (2)^2
16 Simplified Dirac identities
4. Analytical derivation of the same result. The discussion in the preceding section culminated in a geometrical argument designed to illuminate how it comes about that
∆^0 = 0 but truncation engenders ∆ = δ
For comparative purposes I now outline the analytical demonstration of that same fact; the details are not without interest, but the argument as a whole seems to me to lack the “ah-ha!” quality of its geometrical counterpart.
By (18) we have
∆(x − y, t − u) = 12 θ(t − u)θ(&c.) &c. ≡ (t − u)^2 − (x − y)^2
Therefore
∂t∆ = 12 δ(t − u)θ(&c.) + θ(t − u)(t − u)δ(&c.)
∂t∂t∆ = 12 δ′(t − u)θ(&c.) + 2δ(t − u)(t − u)δ(&c.)
- θ(t − u)δ(&c.) + θ(t − u)2(t − u)^2 δ′(&c.)
∂x∆ = − θ(t − u)(x − y)δ(&c.)
∂x∂x∆ = − θ(t − u)δ(&c.) + θ(t − u)2(x − y)^2 δ′(&c.)
giving { ∂ t^2 − ∂ x^2
∆ = 12 δ′(t − u)θ(&c.) + 2δ(t − u)(t − u)δ(&c.)
δ(&c.) + (&c.)δ′(&c.)
where the term that vanishes does so because xδ(x) = 0 =⇒ δ(x) + xδ′(x) = 0. We note in passing that had we omitted the θ-factor from the definition of ∆; i.e., if we were evaluating ∆^0 instead of ∆, we would at this point have achieved ∆^0 = 0.
Our assignment now is to establish that 1 2 δ
′(t − u)θ(&c.) + 2δ(t − u)(t − u)δ(&c.) = δ(t − u)δ(x − y)
which, if we write τ ≡ t − u and ξ ≡ x − y, can be notated
1 2 δ
′(τ )θ(τ 2 − ξ (^2) ) + 2δ(τ )τ δ(τ 2 − ξ (^2) ) ︸ ︷︷ ︸
= δ(τ )δ(ξ)
But = (^12) ∂τ^ ∂
δ(τ )θ(τ 2 − ξ^2 )
+δ(τ )τ δ(τ 2 − ξ^2 )
I will argue that = 0 (19)
Dimensional generalization 17
Assuming, for the moment, the truth of that claim, we want to show that
δ(τ )τ · δ(τ 2 − ξ^2 ) ︸ ︷︷ ︸
= δ(τ )δ(ξ) (∗)
But this is in fact immediate; looking to (11) and taking advantage of simplifications made available by the presence of the δ(τ )-factor, we have
= δ(τ )τ · (^21) τ
δ(ξ − τ ) + δ(ξ + τ )
= δ(τ ) · (^12)
δ(ξ) + δ(ξ)
= δ(τ ) · δ(ξ)
Returning now to the demonstration of (19); if F (τ ) is nice function, then (integrating by parts) we have
∫ ∂ ∂τ
δ(τ )θ(τ 2 − ξ^2 )
F (τ ) dτ = −
δ(τ )θ(τ 2 − ξ^2 )F ′(τ ) dτ
δ(τ )
[
θ(τ + ξ) − θ(τ − ξ)
}]
F ′(τ ) dτ by (12)
δ(τ )F ′(τ ) dτ +
∫ (^) +ξ
−ξ
δ(τ )F ′(τ ) dτ
= 0 all ξ > 0, ∴ also in the limit ξ ↓ 0 because F (τ ) is “nice”
The argument just concluded is notable for its delicacy^13 and its overall improvisatory, ad hoc quality. It is, in my view, not only less illuminating but also less convincing than the geometrical argument summarized in Figures 6–8. It points up the need for a systematic “calculus of distributions,” and lends force to Dirac’s observation that we should (in the absence of such a calculus) avail ourselves of δ-methods only when “it is obvious that no inconsistency can arise.” This, however, is more easily said than done.
5. Relaxation of the assumption that spacetime is 2-dimensional. In §3 we were able to obtain Green’s functions of the homogeneous/inhomogeneous wave equations (^) { ∂ t^2 − ∂^2 x
∆^0 = 0
∂ t^2 − ∂^2 x
∆ = δ
by putting d’Alembert and Heaviside/Dirac in a pot and stirring gently, the notable fact being that no Fourier was called for by the recipe. d’Alembert made critical 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-casual
use of Dirac’s identity xδ(x) = 0; this makes good sense in the context Dirac intended (
xδ(x) · f (x) dx = 0), but can lead to error when (as at (∗)) xδ(x) appears as a factor in a more complex expression.
