Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis by Gerald Kaiser

Author:Gerald Kaiser
Language: eng
Format: epub
Tags: Quantum Gravity,

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5.2. The Multivariate Analytic-Signal Transform

155

/+ and /_ are just the Fourier-Laplace transforms of the restrictions of / to the positive and negative frequencies.

If / is complex-valued, then / + and /_ are independent and the original signal can be recovered from them as

f(x) = ]im\f+(x-iy) + f-(x + iy)]. (3) yio

If / is real-valued, then

Kp) = K-p), (4)

hence / + and /_ are related by reflection,

/+(*) = /_(*), zGflT, (5)

and

f(x) = \im2ftf + (x-iy) = lim2K/_(x + iy). (6)

ylO ylO

When / is real, the function f+(z) is known as the analytic signal associated with f(x). A complex-valued signal would have two independent associated analytic signals /+ and /_. What significance do f± have? For one thing, they are regularizations of /. The above equation states that / is jointly a boundary-value of the pair / + and /_. As such, / may actually be quite singular while remaining the boundary-value of analytic functions. Also, f± provide a kind of "envelope" description of / (see Klauder and Sudarshan [1968], section 1.2). For example, if f(x) = cos ax (a > 0), then f±(z) = | exp(=|=ia,2), so the boundary values are f±(x) = \ exp(=)=iax).

In order to extend the concept of analytic signals to more than one dimension, let us first of all unify the definitions of f+ and /_ by defining

/oo dp6(yp)e-^-^f(p) (7) -oo

for arbitrary x — iy G (E, where 9 is the unit step function, defined by

r0, u < 0

9{u) = < 1/2, u = 0 (8) U, u>0.

5. Quantized Fields

Then we have

(/+(*), y>o

f(z)={±f(x), y = 0 (9)

U-(*), y<o.

[The apparent inconsistency f(x) = \f{x) for y = 0 is due to a mild abuse of notation. It could be removed by redefining f(z) by a factor of 2 or, more correctly but laboriously, rewriting it as (Sf)(z). We prefer the above notation, since the boundary-values f(x) will not actually be used in the phase-space formalism.] Let us define the exponential step function by

0< = 0(-&C) e <, (10) so that our extension is given by

/oo d P 9-^f(p). (11) -oo

The identity

9(u)9{u') = 9{uu')9{u + u') (12) shows that 9^ has the "pseudo-exponential" property

e c e c ' = 9(^c^C')o c+c \ (13)

which will be useful later.

Although this unification of / + and /_ may at first appear to be somewhat artificial, we shall now see that it is actually very natural. Note first of all that for any real u, we have

e( u )eu = — -^—e lTU , (14)

since the contour on the right-hand side may be closed in the lower half-plane when u < 0 and in the upper half-plane when u > 0. For u = 0, the equation states that

m 1 r(r + ,)dr

^ij-oo r 2 + l

i r ir i < 16 >

2tt t' + 1 2

5.2. The Multivariate Analytic-Signal Transform

157

in agreement with our definition, if we interpret the integral as the limit as L —> oo of the integral from — L to L. The exponential step function therefore has the integral

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