{"id":401,"date":"2015-02-07T05:23:05","date_gmt":"2015-02-07T05:23:05","guid":{"rendered":"http:\/\/underthehood.blogwyrm.com\/?p=401"},"modified":"2022-07-28T06:25:37","modified_gmt":"2022-07-28T10:25:37","slug":"one-sided-greens-functions-and-causality","status":"publish","type":"post","link":"https:\/\/underthehood.blogwyrm.com\/?p=401","title":{"rendered":"One-Sided Greens Functions and Causality"},"content":{"rendered":"<p><head><\/p>\n<style>\ntable, th, td { \n    border: 1px solid black;\n    border-collapse: collapse;\n}\nth { text-align: center !important; }\n<\/style>\n<p><\/head><\/p>\n<p>This week we pick up where we left off in the last post and continue probing the structure of the one-sided Greens function $$K(t,\\tau)$$.  While the computations of the previous post can be found in most introductory textbooks, I would be remiss if I didn&#8217;t mention that both the previous post and this one were heavily influenced by two books: Martin Braun&#8217;s &#8216;Differential Equations and Their Applications&#8217; and Larry C. Andrews&#8217; &#8216;Elementary Partial Differential Equations with Boundary Value Problems&#8217;.  <\/p>\n<p>As a recap, we found that a second order inhomogeneous linear ordinary differential equation <\/p>\n<p>\\[ y&#8221;(t) + p(t) y'(t) + q(t) y(t) \\equiv L[y] = f(t) \\; ,  \\]<\/p>\n<p>($$y'(t) = \\frac{d}{dt} y(t)$$) with  boundary conditions<\/p>\n<p>\\[ y(t_0) = y_0 \\; \\; \\&#038; \\; \\;   y'(t_0) = y_0&#8242; \\; \\]<\/p>\n<p>possesses the solution<\/p>\n<p>\\[ y(t) = A y_1(t) + B y_2(t) + y_p(t) \\; ,\\]<\/p>\n<p>where $$y_i$$ are solutions to the homogeneous equation, $$\\{A,B\\}$$ are constants chosen to meet the initial conditions, and $$y_p$$ is the particular solution of the form<\/p>\n<p>\\[ y_p(t) = \\int_{t_0}^{t} \\, d\\tau \\, K(t,\\tau) f(\\tau) \\; . \\]<\/p>\n<p>By historical convention, we call the kernel that propagates the influence of the inhomogeneous term in time (either forward or backward) a one-sided Greens function.  The Wronskian provides the explicit formula <\/p>\n<p>\\[ K(t,\\tau) = \\frac{ \\left| \\begin{array}{cc} y_1(\\tau) &#038; y_2(\\tau\t) \\\\ y_1(t) &#038; y_2(t) \\end{array} \\right| } { W[y_1,y_2](t)}  =  \\frac{ \\left| \\begin{array}{cc} y_1(\\tau) &#038; y_2(\\tau) \\\\ y_1(t) &#038; y_2(t) \\end{array} \\right| } { \\left| \\begin{array}{cc} y_1(\\tau) &#038; y_2(\\tau) \\\\ y_1&#8242;(\\tau) &#038; y_2&#8242;(\\tau) \\end{array} \\right| }  \\; . \\]<\/p>\n<p>for the one-sided Greens function.  Plugging $$t=t_0$$ into the particular solution, gives <\/p>\n<p>\\[ y_p(t_0) = \\int_{t_0}^{t_0} \\, d\\tau \\, K(t_0,\\tau) f(\\tau) =  0 \\]<\/p>\n<p>as the initial datum for $$y_p$$ and <\/p>\n<p>\\[ y_p'(t_0) = K(t_0,t_0) f(t_0) + \\int_{t_0}^{t_0} \\left. \\frac{\\partial}{\\partial t} K(t,\\tau) \\right|_{t=t_0} f(\\tau)  = 0 \\]<\/p>\n<p>for the initial datum for $$y_p&#8217;$$, since the definite integral of any integrand with same lower and upper limits is identically zero and because <\/p>\n<p>\\[ K(t,t) = \\frac{ \\left| \\begin{array}{cc} y_1(t) &#038; y_2(t) \\\\ y_1(t) &#038; y_2(t) \\end{array} \\right| } { W[y_1,y_2](t)}  =  0 \\; . \\]<\/p>\n<p>The initial conditions on the particular solution provide the justification that the constants $$\\{A, B\\}$$ can be chosen to meet the initial conditions or, in other words, the initial values are carried by the homogeneous solutions.<\/p>\n<p>The results for the one-sided Greens function can be extended in four ways that make the practice of handling systems much more convenient.<\/p>\n<h1>Arbitrary Finite Dimensions<\/h1>\n<p>Arbitrary number of dimensions in the original differential equation are handled straightforwardly by the relation that <\/p>\n<p>\\[ K(t,\\tau) = \\frac{ \\left| \\begin{array}{cccc} y_1(\\tau) &#038; y_2(\\tau) &#038; \\cdots &#038; y_n(\\tau) \\\\ y_1&#8242;(\\tau) &#038; y_2&#8242;(\\tau) &#038; \\cdots &#038; y_n'(\\tau) \\\\ \\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\ y_1^{(n-1)}(\\tau) &#038; y_2^{(n-1)}(\\tau) &#038; \\cdots &#038; y_n^{(n-1)} (\\tau) \\\\ y_1(t) &#038; y_2(t) &#038; \\cdots &#038; y_n(t) \\end{array} \\right| } { W[y_1,y_2,\\ldots,y_n](\\tau)} \\]<\/p>\n<p>where the corresponding Wronskian is given by<\/p>\n<p>\\[ W[y_1,y_2,\\cdots,y_n](\\tau) = \\left| \\begin{array}{cccc} y_1(\\tau) &#038; y_2(\\tau) &#038; \\cdots &#038; y_n(\\tau) \\\\ y_1&#8242;(\\tau) &#038; y_2&#8242;(\\tau) &#038; \\cdots &#038; y_n'(\\tau) \\\\ \\vdots &#038; \\vdots &#038; \\ddots &#038; \\vdots \\\\ y_1^{(n-1)}(\\tau) &#038; y_2^{(n-1)}(\\tau) &#038; \\cdots &#038; y_n^{(n-1)} (\\tau) \\\\ y_1^{(n)}(\\tau) &#038; y_2^{(n)}(\\tau) &#038; \\cdots &#038; y_n^{(n)} (\\tau) \\end{array} \\right| \\]<\/p>\n<p>and <\/p>\n<p>\\[ y^{(n)} \\equiv \\frac{d^n y}{d t^n} \\; .\\]<\/p>\n<p>The generation of one-side Greens functions is then a fairly mechanical process once the homogeneous solutions are known and since we are guaranteed that the solutions for initial value problems exist and are unique, the corresponding one-sided Greens functions also exist and are unique. The following is a tabulated set of $$K(t,\\tau)$$s adapted from Andrew&#8217;s book.<\/p>\n<table>\n<caption>One-side Greens Functions &#8211; adapted from Elementary Partial Diff. Eqs. by L. C. Andrews<\/caption>\n<tr>\n<th>Operator<\/th>\n<th>\\[ K(t,\\tau) \\]<\/th>\n<\/tr>\n<tr>\n<td>\\[ D  + b\\]<\/td>\n<td>\\[ e^{-b(t-\\tau)}  \\]<\/td>\n<tr>\n<td>\\[ D^n, \\; n = 2, 3, 4, \\ldots \\]<\/td>\n<td>\\[ \\frac{(t-\\tau)^{n-1}}{(n-1)!} \\]<\/td>\n<\/tr>\n<tr>\n<td>\\[ D^2 + b^2 \\]<\/td>\n<td>\\[ \\frac{1}{b} \\sin b(t-\\tau) \\]<\/td>\n<tr>\n<tr>\n<td>\\[ D^2 &#8211; b^2 \\]<\/td>\n<td>\\[ \\frac{1}{b} \\sinh b(t-\\tau) \\]<\/td>\n<tr>\n<tr>\n<td>\\[ (D-a)(D-b), \\; a \\neq b \\]<\/td>\n<td>\\[ \\frac{1}{a-b} \\left[ e^{a(t-\\tau)} + e^{b(t-\\tau)} \\right] \\]<\/td>\n<tr>\n<tr>\n<td>\\[ (D-a)^n, \\; n = 2, 3, 4, \\ldots \\]<\/td>\n<td>\\[ \\frac{(t-\\tau)^{n-1}}{(n-1)!} e^{a(t-\\tau)} \\]<\/td>\n<tr>\n<tr>\n<td>\\[ D^2 -2 a D + a^2 + b^2 \\]<\/td>\n<td>\\[ \\frac{1}{b} e^{a(t-\\tau)} \\sin b (t-\\tau) \\]<\/td>\n<tr>\n<tr>\n<td>\\[ D^2 -2 a D + a^2 &#8211; b^2 \\]<\/td>\n<td>\\[ \\frac{1}{b} e^{a(t-\\tau)} \\sinh b (t-\\tau) \\]<\/td>\n<tr>\n<tr>\n<td>\\[ t^2 D^2 + t D &#8211;  b^2 \\]<\/td>\n<td>\\[ \\frac{\\tau}{2 b}\\left[ \\left( \\frac{t}{\\tau} \\right)^b &#8211; \\left( \\frac{\\tau}{t} \\right)^b \\right]  \\]<\/td>\n<tr>\n<\/table>\n<h1>Imposing Causality<\/h1>\n<p>The second extension is a little more subtle.  Allow the inhomogenous term $$f(t)$$ to be a delta-function so that the differential equation becomes <\/p>\n<p>\\[ L[y] = \\delta(t-a), \\; \\; y(t_0) = 0, \\; \\; y'(t_0) = 0 \\; .\\]<\/p>\n<p>The particular solution<\/p>\n<p>\\[ y = \\int_{t_0}^t \\, d \\tau \\, K(t,\\tau) \\delta(\\tau &#8211; a) = \\left\\{ \\begin{array}{lc} 0, &#038; t_0 \\leq t < a \\\\ K(t,a), &#038; t \\geq a \\end{array} \\right. \\]\n\nnow represents how the system responds to the unit impulse delivered at time $$t=a$$ by the delta-function. The discontinuous response results from the fact that the system at $$t=a$$ receives a sharp blow that changes its evolution from the unforced evolution it was following before the impulse to a new unforced evolution with new initial conditions at $$t=a$$ that reflect the influence of the impulse.\n\nBy applying a little manipulation to the right-hand side, and allowing $$t_0$$ to recede to infinity, the above result transforms into\n \n \\[ K^+(t,\\tau) = \\left\\{ \\begin{array}{lc} 0, &#038; t_0 \\leq t < \\tau \\\\ K(t,\\tau), &#038; \\tau \\leq t < \\infty \\end{array} \\right.  = \\theta(t-\\tau) K(t,\\tau) \\; ,\\]\n \nwhich is a familiar result from <a href=\"http:\/\/underthehood.blogwyrm.com\/?p=211\">Quantum Evolution &#8211; Part 3<\/a>.  In this derivation, we get an alternative and more mathematically rigorous was of understanding why Heaviside theta function (or step function, if you prefer) enforces causality.  The undecorated one-sided Greens function $$K(t,\\tau)$$ is a mathematical object capable of evolving the system forward or backward in time with equal facility.  The one-sided retarded Greens function $$K^+(t,\\tau)$$ is physically meaningful because it will not evolve the influence of an applied force to a time earlier than the force was applied.<\/p>\n<h1>Recasting in State Space Notation<\/h1>\n<p>An alternative and frequently more insightful approach to solving ordinary differential equations comes in recasting the structure into state space language, in which the differential equation(s) reduce to a set of coupled first order equations of the form<\/p>\n<p>\\[ \\frac{d}{dt} \\bar S = \\bar f(\\bar S; t) \\]<\/p>\n<p><a href=\"http:\/\/underthehood.blogwyrm.com\/?p=171\">Quantum Evolution &#8211; Part 2<\/a> presents this approach applied to the simple harmonic oscillator.  The propagator (or state transition matrix or fundamental matrix) of the system contains the one-sided Greens function as the upper-right portion of its structure.  It is easiest to see that result by working with a second order system with linearly-independent solutions $$y_1$$ and $$y_2$$ and initial conditions $$y(t_0) = y_0$$ and $$y'(t_0) = y&#8217;_0$$.  In analogy with the previous post, the initial conditions can be solved at time $$t_0$$ to yield the expression<\/p>\n<p>\\[ \\left[ \\begin{array}{c} C_1 \\\\ C_2 \\end{array} \\right] = \\frac{1}{W(t_0)} \\left[ \\begin{array}{cc} y_2&#8242; &#038; -y_2 \\\\ -y_1&#8242; &#038; y_1 \\end{array} \\right]_{t_0} \\left[ \\begin{array}{c} y_0 \\\\ y_0&#8242; \\end{array} \\right] \\equiv M_{t_0} \\left[ \\begin{array}{c} y_0 \\\\ y_0&#8242; \\end{array} \\right] \\; , \\]<\/p>\n<p>where the subscript notation $$[]_{t_0}$$ means that all of the expressions in the matrix are evaluated at time $$t_0$$.  Now the arbitrary solution $$y(t)$$ to the homogeneous equation is a linear combination of the independent solutions weighted by the constants just determined<\/p>\n<p>\\[ \\left[ \\begin{array}{c} y(t) \\\\ y'(t) \\end{array} \\right] = \\left[ \\begin{array}{cc} y_1 &#038; y_2 \\\\ y_1&#8242; &#038; y_2&#8242; \\end{array} \\right]_{t} \\left[ \\begin{array}{c} C_1 \\\\ C_2 \\end{array} \\right] \\equiv \\Omega_{t} \\left[ \\begin{array}{c} C_1 \\\\ C_2 \\end{array} \\right] \\equiv \\Omega_{t} M_{t_0} \\left[ \\begin{array}{c} y_0 &#038; y_0&#8242; \\end{array} \\right] \\; .\\]<\/p>\n<p>The propagator, which is formally defined as <\/p>\n<p>\\[ U(t,t_0) = \\frac{\\partial \\bar S(t)}{\\partial \\bar S(t_0) } \\; ,\\]<\/p>\n<p>is easily read off to be<\/p>\n<div style=\"background-color: #d5d5dc; border: solid 1px black;\">\n\\[ U(t,t_0) = \\Omega_{t} M_{t_0} \\; , \\]\n<\/div>\n<p>which, when back-substituting the forms of $$\\Omega_t$$ and $$M_{t_0}$$, gives<\/p>\n<p>\\[ U(t,t_0) =  \\frac{1}{W(t_0)} \\left[ \\begin{array}{cc} y_1 &#038; y_2 \\\\ y_1&#8242; &#038; y_2&#8242; \\end{array} \\right]_{t}  \\left[ \\begin{array}{cc} y_2&#8242; &#038; -y_2 \\\\ -y_1&#8242; &#038; y_1 \\end{array} \\right]_{t_0} \\; .\\]<\/p>\n<p>In state space notation, the inhomogeneous term takes the form $$\\left[ \\begin{array}{c} 0 \\\\ f(t) \\end{array} \\right]$$ and so the relative component of the matrix multiplication is the upper right element, which is<br \/>\n\\[ \\left\\{ U(t,t_0) \\right\\}_{1,2} = \\frac{y_1(t_0) y_2(t) &#8211; y_1(t) y_2(t_0)}{W(t_0)} \\; , \\]<\/p>\n<p>which we recognize as the one-sided Greens function.  Multiplication of the whole propagator by the Heaviside function yields enforces causality and gives the retarded, one-sided Greens function in the $$(1,2)$$ component.<\/p>\n<h1>Using the Fourier Transform<\/h1>\n<p>While all of the machinery discussed above is straightforward to apply, it does involve a lot of steps (e.g., finding the independent solutions, forming the Wronskian, forming the one-sided Greens function, applying causality, etc.).  There is often a faster way to perform all of these steps using the Fourier transform.  This will be illustrated for a simple one-dimensional problem (adapted from <a href=\"http:\/\/www.amazon.com\/Mathematical-Tools-Physics-Dover-Books\/dp\/048648212X\/ref=sr_1_3?s=books&#038;ie=UTF8&#038;qid=1423218451&#038;sr=1-3&#038;keywords=nearing\">&#8216;Mathematical Tools for Physics&#8217;<\/a> by  James Nearing) of a mass moving in a viscous fluid subjected to a time-varying force<\/p>\n<p> \\[ \\frac{dv}{dt} + \\beta v = f(t) \\; ,\\]<\/p>\n<p>where $$\\beta$$ is a constant characterizing the fluid and $$f(t)$$ is the force per unit mass.<\/p>\n<p>We assume that the velocity has a Fourier transform  <\/p>\n<p>\\[ v(t)  = \\int_{-\\infty}^{\\infty} d \\omega \\, V(\\omega) e^{-i\\omega t} \\; \\]<\/p>\n<p>with the corresponding transform pair<\/p>\n<p>\\[ V(\\omega) = \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} dt \\, v(t) e^{+i\\omega t} \\; .\\] <\/p>\n<p>Likewise, the force possesses a Fourier transform<\/p>\n<p>\\[ f(t)  = \\int_{-\\infty}^{\\infty} d \\omega \\,  F(\\omega) e^{-i\\omega t} \\; .\\]<\/p>\n<p>Plugging the transforms into the differential equation yields the algebraic equation<\/p>\n<p>\\[ -i \\omega V(\\omega) + \\beta V(\\omega) = F(\\omega) \\; ,\\]<\/p>\n<p>which is easily solved for $$V(\\omega)$$ and which, when substituted back in, gives the expression for particular solution<\/p>\n<p>\\[ v_p(t) = i \\int_{-\\infty}^{\\infty} d \\omega \\frac{F(\\omega)}{\\omega + i \\beta} e^{-i\\omega t} \\; .\\]<\/p>\n<p>Eliminating $$F(\\omega)$$ by using its transform pair, we find that  <\/p>\n<p>\\[ v_p(t) = \\frac{i}{2 \\pi} \\int_{-\\infty}^{\\infty} d\\tau K(t,\\tau) f(\\tau)  \\]<\/p>\n<p>with the kernel  <\/p>\n<p>\\[ K(t,\\tau) = \\int_{-\\infty}^{\\infty} d \\omega \\frac{e^{-i \\omega (t-\\tau)}}{\\omega + i \\beta} \\; .\\]<\/p>\n<p>This is exactly the form of a one-sided Greens function.  Even more pleasing is the fact that when complex contour integration is used to solve the integral, we discover that causality is already built-in and that what we have obtained is actually  a retarded one-side Green&#8217;s function<\/p>\n<p>\\[ K^+(t,\\tau) = \\left\\{ \\begin{array}{lc} 0 &#038; t < \\tau \\\\ -2 \\pi i e^{-i \\beta(t-\\tau)} &#038; t > \\tau \\end{array} \\right. \\]<\/p>\n<p>Causality results since the pole of the denominator is in the lower half of the complex plane.  The usual semi-circular contour used in Jordan&#8217;s lemma must be in the upper half-plane when $$t < \\tau$$, in which case no poles are contained and no residue exists.  When $$t > \\tau$$, the semi-circle, which must be in the lower-half plane, and surrounds the pole at $$\\omega = &#8211; i \\beta$$, giving a non-zero residue.<\/p>\n<p>The final form of the particular solution is<\/p>\n<p>\\[ v_p(t) = \\int_{-\\infty}^t d \\tau e^{-\\beta (t-\\tau)} f(\\tau) \\]<\/p>\n<p>which is the same result we would have received from using the one-sided Greens function for the operator $$D + \\beta$$ shown in the table above.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This week we pick up where we left off in the last post and continue probing the structure of the one-sided Greens function $$K(t,\\tau)$$. While the computations of the previous&#8230; <a class=\"read-more-button\" href=\"https:\/\/underthehood.blogwyrm.com\/?p=401\">Read more &gt;<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[19,7,10,18,17],"class_list":["post-401","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-causality","tag-evolution","tag-greens-function","tag-state-space","tag-wronskian"],"_links":{"self":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/401","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=401"}],"version-history":[{"count":24,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/401\/revisions"}],"predecessor-version":[{"id":2175,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/401\/revisions\/2175"}],"wp:attachment":[{"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=401"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=401"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/underthehood.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}