In Many Different . . . Laplace Equation General Case of Linear . . . Additional Conditions . . . Derivation of Scale-Invariance: . . . Louisville-Bratu-Gelfand Shift-Invariance: . . . From Laplace . . . Equation from Shift- or How to Describe Non- . . . Which Function f ( ϕ ) . . . Scale-Invariance Home Page Leobardo Valera, Martine Ceberio, and Vladik Kreinovich Title Page ◭◭ ◮◮ Department of Computer Science, University of Texas at El Paso El Paso, Texas 79968, USA ◭ ◮ leobardovalera@gmail.com, mceberio@utep.edu, vladik@utep.edu Page 1 of 32 Go Back Full Screen Close Quit
In Many Different . . . Laplace Equation 1. In Many Different Situations, We Have the Ex- General Case of Linear . . . act Same Louisville-Bratu-Gelfand Equation Additional Conditions . . . • In many different physical situations, we encounter the Scale-Invariance: . . . same differential equation Shift-Invariance: . . . From Laplace . . . ∇ 2 ϕ = c · exp( a · ϕ ) . How to Describe Non- . . . • This equation – known as Louisville-Bratu-Gelfand equa- Which Function f ( ϕ ) . . . Home Page tion – appears: Title Page – in the analysis of explosions, ◭◭ ◮◮ – in the study of combustion, – in astrophysics (to describe the matter distribution ◭ ◮ in a nebula), Page 2 of 32 – in electrodynamics – to describe the electric space Go Back charge around a glowing wire – and Full Screen – in many other applications areas. Close Quit
In Many Different . . . Laplace Equation 2. Challenge General Case of Linear . . . • The same equation appears in many different situa- Additional Conditions . . . tions. Scale-Invariance: . . . Shift-Invariance: . . . • This seems to indicate that this equation should not From Laplace . . . depend on any specific physical process. How to Describe Non- . . . • It should be possible to derive it from general princi- Which Function f ( ϕ ) . . . ples. Home Page • In this paper, we show that this equation can be nat- Title Page urally derived from basic symmetry requirements. ◭◭ ◮◮ ◭ ◮ Page 3 of 32 Go Back Full Screen Close Quit
In Many Different . . . Laplace Equation 3. Laplace Equation General Case of Linear . . . • The simplest form of our equation is when c = 0. Additional Conditions . . . Scale-Invariance: . . . • In this case, we get a linear equation ∇ 2 ϕ = 0. Shift-Invariance: . . . • This equation is known as the Laplace equation; so: From Laplace . . . – in order to understand where our equation comes How to Describe Non- . . . from, Which Function f ( ϕ ) . . . Home Page – let us first recall where the Laplace equation comes from. Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 32 Go Back Full Screen Close Quit
In Many Different . . . Laplace Equation 4. Scalar Fields Are Ubiquitous General Case of Linear . . . • To describe the state of the world, we need to describe Additional Conditions . . . the values of all physical quantities at all locations. Scale-Invariance: . . . Shift-Invariance: . . . • In physics, the dependence ϕ ( x ) of a physical quantity From Laplace . . . ϕ on the location x is known as a field . How to Describe Non- . . . • Typical examples are components of an electric or mag- Which Function f ( ϕ ) . . . netic fields, gravity field, etc. Home Page • In general, at each location x , there are many different Title Page physical fields. ◭◭ ◮◮ • In some cases, several fields are strong enough to affect ◭ ◮ the situation. Page 5 of 32 • So, we need to take several fields into account. Go Back • However, in many practical situations, only one field is Full Screen strong enough. Close Quit
In Many Different . . . Laplace Equation 5. Scalar Fields Are Ubiquitous (cont-d) General Case of Linear . . . • For example: Additional Conditions . . . Scale-Invariance: . . . – when we analyze the motion of celestial bodies, Shift-Invariance: . . . – we can safely ignore all the fields except for gravity. From Laplace . . . • Similarly, if we analyze electric circuits, we can safely How to Describe Non- . . . ignore all the fields but the electromagnetic field. Which Function f ( ϕ ) . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 32 Go Back Full Screen Close Quit
In Many Different . . . Laplace Equation 6. Case of Weak Fields General Case of Linear . . . • In general, equations describing fields are non-linear. Additional Conditions . . . Scale-Invariance: . . . • However, in many real-life situations, fields are weak. Shift-Invariance: . . . • In this case, we can safely ignore quadratic and higher From Laplace . . . order terms in terms of ϕ and consider linear equations. How to Describe Non- . . . Which Function f ( ϕ ) . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 32 Go Back Full Screen Close Quit
In Many Different . . . Laplace Equation 7. General Case of Linear Equations General Case of Linear . . . • In physics, usually, we consider second order differen- Additional Conditions . . . tial equations, i.e., equations that depend: Scale-Invariance: . . . Shift-Invariance: . . . – on the field ϕ , From Laplace . . . = ∂ϕ def – on its first order partial derivatives ϕ ,i and How to Describe Non- . . . ∂x i Which Function f ( ϕ ) . . . ∂ 2 ϕ def – on its second order derivatives ϕ ,ij = . Home Page ∂x i ∂x j Title Page • The general linear equation containing these terms has the form ◭◭ ◮◮ 3 3 3 ◭ ◮ � � � a ij · ϕ ,ij + a i · ϕ ,i + a · ϕ = 0 . Page 8 of 32 i =1 j =1 i =1 Go Back Full Screen Close Quit
In Many Different . . . Laplace Equation 8. Rotation-Invariance General Case of Linear . . . • In general, physics does not change if we simply rotate Additional Conditions . . . the coordinate system. Scale-Invariance: . . . Shift-Invariance: . . . • Thus, it is reasonable to require that the system be From Laplace . . . invariant with respect to arbitrary rotations. How to Describe Non- . . . • This requirement eliminates the terms proportional to Which Function f ( ϕ ) . . . the first derivatives ϕ ,i . Home Page • Otherwise, we have a selected vector a i and thus, an Title Page expression which is not rotation-invariant. ◭◭ ◮◮ • Similarly, we cannot have different eigenvector of the ◭ ◮ matrix a ij – this would violate rotation-invariance. Page 9 of 32 • Thus, this matrix must be proportional to the unit Go Back matrix with components δ ij = 1 if i = j else 0. Full Screen Close Quit
In Many Different . . . Laplace Equation 9. Rotation-Invariance (cont-d) General Case of Linear . . . • So, a ij = a 0 · δ ij for some a 0 , and the above equation Additional Conditions . . . 3 Scale-Invariance: . . . takes the form a 0 · � ϕ ,ii + a · ϕ = 0 . Shift-Invariance: . . . i =1 From Laplace . . . • Dividing both sides by a 0 and taking into account that 3 How to Describe Non- . . . ϕ ,ii = ∇ 2 ϕ , we get the equation � Which Function f ( ϕ ) . . . i =1 Home Page def ∇ 2 ϕ + m · ϕ = 0 , where m = a/a 0 . Title Page • This equation is indeed the general physics equation ◭◭ ◮◮ for a weak scalar field. ◭ ◮ • The case of m = 0 corresponds to electromagnetic field Page 10 of 32 or gravitational field. Go Back Full Screen Close Quit
In Many Different . . . Laplace Equation 10. Rotation-Invariance (cont-d) General Case of Linear . . . • More generally, m = 0 corresponds to any field whose Additional Conditions . . . quanta have zero rest mass. Scale-Invariance: . . . Shift-Invariance: . . . • Example: photons or gravitons, quanta of the above From Laplace . . . fields. How to Describe Non- . . . • In the general case, when the quanta have non-zero rest Which Function f ( ϕ ) . . . mass, we get a more general equation with m � = 0. Home Page • Example: strong interactions whose quanta are π -mesons. Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 32 Go Back Full Screen Close Quit
In Many Different . . . Laplace Equation 11. Additional Conditions Are Needed to Pinpoint General Case of Linear . . . Laplace Equation. Additional Conditions . . . • Can we explain why: Scale-Invariance: . . . Shift-Invariance: . . . – out of all possible equations of type, From Laplace . . . – Laplace equation – corresponding to m = 0 – is the How to Describe Non- . . . most frequent? Which Function f ( ϕ ) . . . • We need to use additional conditions. Home Page • As such conditions, we will use the fundamental no- Title Page tions of scale- and shift-invariance. ◭◭ ◮◮ ◭ ◮ Page 12 of 32 Go Back Full Screen Close Quit
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