# Equations

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## INTRODUCTION

I start this page with a simple equation with Walter Lewin at MIT May 16 2011 about the frequence of a pendulum.

# INDEX

## Partial derivative

The term “Partial derivative” has confused me with its new symbol ∂(curly d).

Let say  you have  a function  that depends on  e.g. 2 variables or is a composite function h(x) like h(x) = g(f(x)) with a inner function  f(x)(compare to a nested function in programming). If you want derivate this function, then you have to make 2 partial derivations with the so called chain rule. Every partial derivation is a partial derivative that is denoted with a curly d that has the  character (Unicode: U+2202). It is a stylized cursive/curly d.

A example is

### Chain rule

To write a equation for a 2 frame system, we need the chain rule that tells how to derivate a external function with a inner function.

In calculus, the chain rule is a formula to compute the derivative of a composite function. That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their compositef ∘ g — the function which maps x to — in terms of the derivatives of f and g and the product of functions as follows:

I found 3 applications of this rule: from  NancyPi Youtube

1 example from Khan

### Derivation with Leibnitz notation

Derviation with the chain rule with Leibnitz notation is used in the derivation of the Metric tensor in EFE. I found a good teacher in the video below.

For a function

here isf(x)  with a inner function g(x), one can write the chain rule in Lagrange’s notation, as follows:

The chain rule may also be rewritten in Leibniz’s notation in the following way. If a variable Φ depends on the variable x, which itself depends on the variable y (i.e., y and z are dependent variables), thenΦ, via the intermediate variable of x, depends on y as well. In which case, the chain rule states that

(Source: Wiki  )

Here is the chain rule made using Leibnitz notation, explained:

I took this screenshot that describes the method used in the video. Now I can proceed with DrPhysics metric tensor derivation. 🙂

## Newtons second law of rotation

Khan Academy explains it well in this video

The rotational version of Newton’s law

is

where denotes the angular acceleration. As in the previous section, is torque (tangential force times a moment arm ), and is the mass moment of inertia. Thus, the net applied torque equals the time derivative of angular momentum , just as force equals the time-derivative of linear momentum Read more at pccrma.stanford

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### Trigonometry

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy.[1] In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE), who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. (source: Wiki )

The sinus(x) functions can be calculated by a a computer program with the use of factorialss in the following formula:

sin(x) = x – x^3 / 3! + x^5 / 5! – x^7 / 7! + x^9 / 9!

Formula taken from https://www.mathworks.com where a program algoritm is discussed. I wonder who developed this solution.

it is called the Taylor series. Taylor’s series are named after Brook Taylor who introduced them in 1715.

For sin(x) it is as follows.

(  Image source: Wiki  )

## For cos(x) it is as follows:

(  Image source: Wiki  )

### L functions

Studying anf trying to understand Riemann’s hypothesis, I came up looking at the L functions.

L-functions were introduced by Dirichlet in the early 19th century “to study prime numbers in arithmetic progressions. Twenty years later Riemann took a major step forward by demonstrating that how the study of these L-functions as functions of a complex variable, could hold the key to the deep mysteries about the distribution of prime numbers. In the same paper Riemann introduced his famous hypothesis which is now considered the ‘holy grail’ of mathematics.” (Read more at https://www.youtube.com/watch?v=A7Q3mLTq8M4 )

“In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. The Riemann zeta function is an example of an L-function, and one important result involving L-functions is the Riemann hypothesis and its generalization.”

In mathematics, a Dirichlet series is any series of the form

where s is complex, and  is a complex sequence. It is a special case of general Dirichlet series.
In mathematics, a Dirichlet L-series is a function of the form

Here χ is a Dirichlet character (In number theoryDirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of . )

and s a complex variable with real part greater than 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ).

( Wiki )

## The Euler´s identity,

that the physicist Richard Feynman called “our jewl”,  is derived from the Taylor series of sin (x) and cos(x) .

( Image source: www.livescience.com )

## The period of the pendulum

Walter Lewin demonstrates in this video , the validity of the pendulum Equation. Read more at www.acs.psu.edu/

derivation as described by www.acs.psu.edu

A simple pendulum consists of a ball (point-mass) m hanging from a (massless) string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. By applying Newton’s secont law for rotational systems, the equation of motion for the pendulum may be obtained .

Equation with rearrangedment belows

If the amplitude of angular displacement is small enough, so the small angle approximation ($\sin\theta\approx\theta$) holds true, then the equation of motion reduces to the equation of simple harmonic motion

The simple harmonic solution is

where is the initial angular displacement, and the natural frequency of the motion. The period of this system (time for one oscillation) is

T=la

=

-mg sin 0 L=ml

m2 do

d+2

and rearranged as

de 9

dt2 +

sinA=0

If the amplitude of angular displacement is small enough, so the small angle approximation ($\sin\thetalapprox\theta$) holds true, then the equation of motion reduces to the equation of simple harmonic motion

m

I

mg sino

d20

g

dt2 + 10=0

mg cose

mg

The simple harmonic solution is

e(t) = 0, cos(wt),

where an is the initial angular displacement, and w= 1 g/L the natural frequency of the motion. The period of this sytem (time for one oscillation) is

## Maxwell’s equations

Maxwell’s equations describe how electric and magnetic fields are generated by chargescurrents, and changes of the fields. An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in a vacuum.

Issues in the video:

• Quantum mechanics requires understanding of waves.
• The magnetic field is a vector fields
• Maxwell equations tells how electric and magnetic fields change each others
• Horizontal polarized waveE=Acos(2π,f,t+Øx) Ι→〉+0Ι↑〉
where 0Ι↑〉 is the unit vector in vertical direction
• Vertically polarized wave: E=0 Ι→ + Ay cos(2π,f,t+Øx)
where 0Ι→〉 is the vector in horizontal direction

where their videos are financed by course payments about €5.54 /month.

.

A introduction part 1 of electromagnetism is made here:

As Flux density = B =  ϕ/A ( Wbm-2 )

so Flux = field strength per unit area ϕ= BA (unit Wb)

B (unit tesla)

Flux linkage  ( =BAn ) in a coil with N coils

or
where  s the voltage across the device
Maxwell came up with the right hand rule (or cork screw rule) to find out how magnetic field goes around a piece of wire with a current in it.. Khan academy presents this law here:
Key notations:
is a surface integral over the surface Σ,
• Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated.
•  the magnetic fieldB, a pseudovector field, each generally having a time and location dependence.

The sources are:

• the total electric charge density (total charge per unit volume), ρ, and
• the total electric current density (total current per unit area), J.

The universal constants appearing in the equations (the first two ones explicitly only in the SI units formulation) are:

In the differential equations,

• the nabla symbol, denotes the three-dimensional gradient operator, del,
• the ∇⋅ symbol (pronounced “del dot”) denotes the divergence operator,
• the ∇× symbol (pronounced “del cross”) denotes the curl operator.

## Laplace equation

Laplace’s equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

∇ 2 f = 0 or Δ f = 0 , {\displaystyle \nabla ^{2}\!f=0\qquad {\mbox{or}}\qquad \Delta f=0,}

where Δ = ∇ ⋅ ∇ = ∇ 2 {\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}} is the Laplace operator,[note 1] ∇ ⋅ {\displaystyle \nabla \cdot } is the divergence operator (also symbolized “div”), ∇ {\displaystyle \nabla } is the gradient operator (also symbolized “grad”), and f ( x , y , z ) {\displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.

If the right-hand side is specified as a given function, h ( x , y , z ) {\displaystyle h(x,y,z)} , we have

Δ f = h . {\displaystyle \Delta f=h.}

This is called Poisson’s equation, a generalization of Laplace’s equation. Laplace’s equation and Poisson’s equation are the simplest examples of elliptic partial differential equations.

With Laplace equation we get:

### Laplace’s tidal equations

the dynamic theory of tides, developed by Pierre-Simon Laplace in 1775,[9] describes the ocean’s real reaction to tidal forces. Laplace’s theory of ocean tides took into account friction, resonance and natural periods of ocean basins. It predicted the large amphidromic systems in the world’s ocean basins and explains the oceanic tides that are actually observed. Laplace obtained these equations by simplifying the fluid dynamic equations, but they can also be derived from energy integrals via Lagrange’s equation. ( Wiki )

For a fluid sheet of average thickness D, the vertical tidal elevation ζ, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy  Laplace’s tidal equations:[28]

where Ω is the angular frequency of the planet’s rotation, g is the planet’s gravitational acceleration at the mean ocean surface, a is the planetary radius, and U is the external gravitational tidal-forcing potential. ( Wiki )

.

### Eulers formula

Eulers formula was used by Riemann to derivate his Zeta function described in a separate page.

(  Image source: Wiki  )

The Taylor series for the exponential function ex at a = 0 is

(  Image source: Wiki  )

( Image source: Khan video )

( Image source: Khan video )

### Euler’s identity

In this video Mark Newman describes beatifully the relation between sinusoids and Euler number in Eulers formula:

If I am not wrong….

If e= -1

then e= i2

i sqr(e)

!EULER’S IDENTITY A MATHEMATICAL PROOF FOR THE EXISTENCE OF GOD! !in https://www.academia.edu/217151/Eulers_Identity

## The schrödinger equation derivation

In its simplest form, the Schroedinger equation looks like this

(Source: screenshots from Parth video )

where i  is the imaginary number, thethe square root of -1

•  ℏ  is related to the Plank’s Constant as
• Ĥ  is what we call Hamiltonian.
• ψ represent wave functions in quantum mechanics.
• the line and pointed bracket around theψ (PSI) is indicating that it is a a quantum state.  The quantum state is a mathematical description of a quantum system,  e.g. a electron near the nucleus.

The scroedinger equation Calculates the probability for a particle (e.g. electron) to be in a certain position. Iit s shared by Brian Green in this World Science Festival video.

Unnikrishnan Menon,  derived it for us  in Quora.

I hope he don’t mind I share his loooong derivation here. He writes:

It is all about finding the different energies that a particle can have.

We’re looking at things that can have more than one answer! You might have studied in high school that atoms have energy levels. The Schrödinger’s Wave Equation lets us calculate what these energies are.

It is defined as:

It turns out, talking about velocity isn’t very useful. So, we change this equation to make it depend on momentum p

We can think of particles as waves, at least at tiny scales where we need to use Quantum Mechanics!

To help us move between these 2 ways of thinking about matter, we can use De Broglie’s Equation

where  is the wavelength is the Planck constant is the momentum is the rest mass,   is the velocity and  is the speed of light in a vacuum.”

This theory set the basis of wave mechanics. It was supported by Einstein, confirmed by the electron diffraction experiments of G P Thomson and Davisson and Germer, and generalized by the work of Schrödinger. ( Wiki )

Now we don’t see matter around us in our everyday lives to be behaving like waves because Plank’s Constant h is absolutely tiny! ( h=6.62×10−34m2Kg/s )

But hold on! De Broglie’s Equation is useful when we are dealing with minute particles like protons and electrons 🙂

We also have something called ℏ
It’s is related to the Plank’s Constant as

Now, it is going to be useful to talk about  and the wave number instead of h and λ

Where

is the wave number. We did to avoid having  running around in our equations. It just turns out to be easier to work with!
We also know that k is change depending upon the energy. Since k depends only on λ , the factors that can affect λ can also affect k . In this case, it is Energy!

We use Wave Functions to encode information about waves. These are just equations with a few special properties. They look something like this…

This is just another way to write sines and cosines. (Waves are just combinations of sines and cosines!)

Here’s the reason for writing this in this weird format…

This type of problem is called an Eigen Value Problem.

is called an Eigen Function.

is an Operator.

Eigen Function of operators are functions which return themselves and a new Eigen Value after the operator is used on them.

As you might have guessed by now, on differentiating this we get the same thing back multiplied by some constants (differential of the argument)

Let’s do this again!

Here, −k2 is the new Eigen Value returned (that’s the extra bit)

Now let’s make a Kinetic Energy Operator (we want the energy EE and we are assuming a low potential)…

We know there are gonna be a bunch of answers so we are going to use our Wave Function to generate those answers and we also have a way of writing k2

So on putting it all together we get…

Notice it is partial derivative instead of the usual one. This just means that there is more than one variable in the function we treat as a constant when differentiating.

What we just did is valid for a one dimensional particle that is time independent and is not in a potential.

Let’s generalize this equation a little…

For a photon,

We’re now gonna assume it to be true for particles. It turns out to be a good assumption…

We know that there is gonna be a set of energies so we want to know the Eigenvalue equation.

This time we’ll find an E by generating an ω

Let’s look at the wave function to do that…

As you might have guessed it by now, we’re gonna take the derivative with respect to time to pull it out front!

To account for the minus sign(-) we will simply use -i.i=1

And that’s same as the equation we had before…

Please note that this equation is only for one spatial dimension! But we need to consider all 3 dimensions. The three dimensions in Cartesian Coordinate System are orthogonal to each other. That means, they will not interfere with each other.

Each of these contributes to the total energy.

So, in three dimensions, the time dependent wave equation becomes…

And the time independent version becomes…

This can more simplified as…

Where is what we call Nabla.

To include Potential(V) we modify it a bit further like this…

where Ĥ  is what we call Hamiltonian.

is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system (this addition is the total energy of the system in most of the cases under analysis ( Wiki )

So, the full Schrödinger’s Wave Equation becomes…

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### The classical wave function

I shared above The Schrödinger equation derivation.

The classical order wave function is a one dimension (linear) partial derivative function that has no relativity nor quantum mechanics included.

Parth shares a very good presentation of it at

Possible solution  of the wave equation as explained by Parth:

### general relativity field equation (EFE)

First part copied from my page at www.kinberg.net/wordpress/stellan/einsteins-field-equations-of-general-relativity/

As is correctly explained in Wikipedia

I found this Youtube that explains the general relativity field equation. I like it because of the good graphic representations.

As with the tensor youtube made by Dan Fleisch, this made by Eugene Khutoryansky show graphically how a tensor change depending on curvature of space. Very well done video.

## .

### Einstein field equation (EFE) tensors

I copied this part to my tensor page.
This is Einsteins field equation (EFE) in simplified form. It is called Einstein Field equations as it is really 16 equations  depending on the value of the indeces (μν) for g and T in the equation. 6 are duplicates. (read more below)

(Source: Dr Physics A Youtube )

This is a very good video that I recommend where . DrPhysicsA explains some basics and derives the equation.

• How you can feel gravitation and acceleration force
• How light bends in gravitation fields.
• 5:33 Proven at a solar eclipse
• 5:50 Why Newtons gravitation law does not apply to (0 mass) photons.
•  6:42 All forms of movement are represented by movement in curved spacetime
• Trampoline metaphore
• 9:00 What is spacetime?
• 11:00 Not only one equation as it looks like.
The indeces μν represent the spacetime 4×4. 6 are duplicates = 10 EFE
• 13:15. “Everything on the left hand of EFE refers to curvature of spacetime. Everything on the right hand side is to do with mass and energy. What Einstein field equation basically say is that
‘mass tells  space-time how to curve
and curved space-time tells mass how to move’

Deriving EFE.
• 15:15 A Where I am on a bumpy field
• 16;13 How does my height change as I move on the field
• 14:50 metric tensor
• 24:35 relableing the coordinates.
• 19:17
• 32:05 Tensor – relationship between two vectors.
• 42:17  Contravariant and covariant transformation
• 46:15 Kronecker delta
• 49:45 Metric tensor – correct pythagoras equation on a curved space.
• 53:14 Christophel symbol

### Pythagoras and vectors

I were confused of his using ϕ in the same way as dh where h=height in the field. I understand that dϕxis a movement in the field in parallell to the x ax.

With Pythagoras we get a formula for 2 dimensional vectors like ds in the printscreen above.

The term “Partial derivative” has confused me with its new symbol . With this video I understand that a partial derivative is just a derivative that is not the only one to describe the movement. A derivative with respect to one of several variables. To denote a partial derivative  character (Unicode: U+2202) is used. It is a stylized cursive/curly d.

relableing the coordinate system

This is necessary to be able to have more coordinates as in e.g. a 11 dimension system.If you are not used to it you get rather confused with the following calculations. So I keep this screenshot here as a memory backup. The y-coordinate will still be used but for another set of coordinates/frame of reference. x0 is used for time coordinate.

With the new coordinate system

can be rewritten as

If we have2 reference systems, then we have

Chain rule

In calculus, the chain rule is a formula to compute the derivative of a composite function. That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their composite fg — the function which maps x to f ( g ( x ) ) {\displaystyle f(g(x))} — in terms of the derivatives of f and g and the product of functions as follows:

Alternatively, by letting F = fg (equiv., F(x) = f(g(x)) for all x), one can also write the chain rule in Lagrange’s notation, as follows:

F ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x ) .  F'(x)=f'(g(x))g'(x).}
Wiki  )
The chain rule may also be rewritten in Leibniz’s notation in the following way. If a variable z depends on the variable y, which itself depends on the variable x (i.e., y and z are dependent variables), then z, via the intermediate variable of y, depends on x as well. In which case, the chain rule states that:

d z d x = d z d y ⋅ d y d x . {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}.}

More precisely, to indicate the point each derivative is evaluated at, d z d x | x = d z d y | y ( x ) ⋅ d y d x | x  .

To write a equation for a 2 frame system we need the chain rule that tells how to derivate a function in  a outer function.

I found 3 applications of this rule:

3 from  NancyPi

1 example from Khan

So with the chain rule,  we can describe the position in a 2 dimensional system as

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## Introduction to Information Theory

I dont know why Edward Witten choose to start with a short introduction to Communication theory (the Shannon theory) at “Theoretical Physics 2018: From Qubits to Spacetime”

I presume it is important to know about this, to understand the theory of Quantum mechanics and aspects of General Relativity as he says to continue with at the conference.
I decided to take a look at C.E. Shannons book from 1948 ” A mathematical theory of Communication

Shannon was an American mathematician, electrical engineer, and cryptographer known as “the father of information theory“. He wrote also “Theoretical Genetics.”[12]

A mathematical theory of Communication – Shannon theory

Khan Academy has a great introduction to Shannons theory in this video: (in youtube settings choose your subtitle language).
the video with English text is here.

C.E. Shannons book from 1948 ” A mathematical theory of Communication“. The instructor in this Khan video says among others:
“Claude Elwood Shannon developed theory about cryptography. Then Claude Shannon demonstrated how to generate “English looking” text using  Markov chains.
Bernouilli. Weak law of large number says: As the number of trials increases, the expected ratio value will converge on the actual underlying ratio. He refined the idea of expectation. If observations of all the events will be continued for the entire infinity it will be noticed that “Everything in the world is governed by precise ratios and a constant law of change.”
The binomial distribution appears to be an ideal form as it kept appearing everywhere anytime  you looked at a the variation of a large number of random trials. It seems the average fate of these events are somehow predetermined, known today as  “The central limit theorem”
Most of things in the physical world are clearly dependent on prior  outcomes. We talk about dependent events or dependent variables.
Markov prooved that independent and dependent events can converge on predictable distributions.
One of the most famous application of Markov chains was published by Claude Shannon.”

## What Every Physicist Should Know About String Theory

I listened to  Edward Witten in this Youtube:

The slide texts are very difficult to read and Witten is very fast and you get little time to think over it.

Here are the slide texts  he presented so you can hopefully read these at your pace and maybe also get it satisfactory translated to your own language:

Slide 1.I am going to try today to explain the minimum that any physicist might want to know about string theory. I will try to explain answers to a couple of basic questions.

• How does string theory generalize standard quantum field theory?
• And why does string theory force us to unify General Relativity with the other forces of nature, while standard quantum field theory makes it so difficult to incorporate General Relativity?
• Why are there no ultraviolet divergences?
• And what happens to Einstein’s conception of spacetime?

I thought .that explaining these matters is possibly suitable for a session devoted to the centennial of General Relativity.

Slide 2. Anyone who has studied physics is familiar with the fact that while physics – like history – does not precisely repeat itself, it does rhyme, with similar structures at different scales of lengths and energies. We will begin today with one of those rhymes – an analogy between the problem of quantum gravity and the theory of a single particle.

Slide 3 04:32
Even though we do not really understand it, quantum gravity is supposed to be some sort of theory in which, at least from a macroscopic point of view, we average, in a quantum mechanical sense, over all possible spacetime geometries….

(We do not know to what extent this description is valid microscopically.) The averaging is done, in the simplest case, with a weight factor exp(-1) (1 will write this in Euclidean signature) where I is the Einstein-Hilbert action

with R being the curvature scalar and (lambda) the cosmological constant. We could add matter fields, but we don’t seem to have to.

Slide 4 05:20

Let us try to make a theory like this in spacetime dimension 1, rather than 4. There are not many options for a 1-manifold

In contrast to the 4d case, there is no Riemann curvature tensor in 1 dimension so there is no close analog of the Einstein-Hilbert action.

Slide 5.

Even there is no l Integral[(root(g)R] to add to the action, we can still make a nontrivial theory of “quantum gravity,” that is a fluctuating metric tensor, coupled to matter. Let us take the matter to consist of some scalar fields Xi j = 1…..D (matter fields). The most obvious action is

where g = (gut) is a 1×1 metric tensor and I have written ma2 / 2 instead of lambda

Slide 6 (repeating slide 5)

If we introduce the “canonical momentum Pi=dXi/dt

then the “Einstein field equation” is just

In other words, the wavefunction ω(X) should obey the corresponding differential equation

Slide 7 (8:32)

This is a familiar equation – the relativistic Klein-Gordon equation in D dimensions – but in Euclidean signature. If we want to give this fact a sensible physical interpretation, we should reverse the sign of the action for one of the scalar fields X; so that the action becomes

Now the equation obeyed by the wavefunction is a Klein-Gordon equation in Lorentz signature:

Slide 8  9:07

So we have found an exactly soluble theory of quantum gravity in one dimension that describes a spin 0 particle of mass m propagating in D-dimensional Minkowski spacetime. Actually, we can replace Minkowski spacetime by any D-dimensional spacetime M with a Lorentz (or Euclidean) signature metric GIJ. the action being then

The equation obeyed by the wavefunction is now a Klein-Gordon equation on our spacetime M:

This is the massive Klein-Gordon in curved spacetime

Slide 9  10:23

Just to make things more familiar, let us go back to the case of flat spacetime, and I will abbreviate G PIPJ, as P2 (To avoid keeping track of some factors of i. I will also write formulas in Euclidean signature.) Let us calculate the amplitude for a particle to start at a point x in spacetime and end at another point y.

Part of the process of evaluating the path integral in a quantum gravity theory is to integrate over the metric on the one-manifold, modulo diffeomorphisms. But up to diffeomorphism, this one-manifold has only one invariant, the total length 1, which we will interpret as the elapsed proper time.

Slide 10 12:25 “Keeping To fixed”

For a given Tao we can take the 1-metric to be just gtt = 1 where
0 <t<T (As a minor shortcut, I will take Euclidean signature on the l-manifold. “as described by Feinman 50 years ago” Now on this l-manifold, we have to integrate over all paths X(t) that start at x at t = 0 and end at y at t = tao.

Slide 11 14:17

For a given tao we can take the l-metric to be just gtt = 1 where 0<=t<=tao (As a minor shortcut. I will take Euclidean signature on the l-manifold.) Now on this l-manifold, we have to integrate over all paths X(t) that start at x att 0 and end at y at t = This is the basic Feynman integral of quantum mechanics with the Hamiltonian being H P2 + m and according to Feynman, the result is the matrix element of exp(-taoxH) (H= Hamiltonian):

But we have to remember to do the “gravitational” part of the path integral, which in the present context means to integrate over

Slide 12 14:17

Thus the complete path integral for our problem – integrating over all metrics Sur(t) and all paths X(t) with the given endpoints, modulo diffeomorphisms – gives

This is the standard Feynman propagator in Euclidean signature, and an analogous derivation in Lorentz signature (for both the spacetime M and the particle worldline) gives the correct Lorentz signature Feynman propagator, with the iE

Slide 12 15:15

So we have interpreted a free particle in D-dimensional spacetime in terms of 1-dimensional quantum gravity. How can we include interactions? There is actually a perfectly natural way to do this. There are not a lot of smooth l-manifolds, but there is a large supply of singular l-manifolds in the form of graphs.

Our “quantum gravity” action makes sense on such a graph. We just take the same action that we used before, summed over all of the line segments that make up the graph

Slide 13 16:02

Now to do the quantum gravity path integral, we have to integrate over all metrics on the graph, up to diffeomorphism. The only invariants are the total lengths or “proper times” of each of the segments:

(I did not label all of them.)

Slide 14 17:14

To integrate over the paths, we just observe that if we specify the positions y….. y4 at all the vertices (and therefore on each end of each line segment)

then the computation we have to do on each line segment is the same as before and gives the Feynman propagator. Integrating over the y; will just impose momentum conservation at vertices, and we arrive at Feynman’s recipe to compute the amplitude attached to a graph: a Feynman propagator for each line, and an integration over all momenta subject to momentum conservation.

Slide 15 19:07

We have arrived at one of nature’s rhymes: if we imitate in one dimension what we would expect to do in D = 4 dimensions to describe quantum gravity, we arrive at something that is certainly important in physics, namely ordinary quantum field theory in a possibly curved spacetime. In the example that I gave, the “ordinary quantum field theory’ is scalar 03 theory, because of the particular matter system we started with and assuming we take the graphs to have cubic vertices. Quartic vertices (for instance) would give 4 theory, and a different matter system would give fields of different spins. So many or maybe all QFT’s in D dimensions can be derived in this sense from quantum gravity in 1 dimension.

Slide 16 19:07
here is actually a much more perfect rhyme if we repeat this in two dimensions, that is for a string instead of a particle. One thing we immediately run into is that a two-manifold E can be curvedthe integral over 2d metrics promises to not be trivial at all

## conclusions

this is a tough issue to deal with. The last theory that mathematicians are working with are the Theory M. I decided to start a new issue “Introduction to Information theory

## A agnostic pluralist seeker

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