Maple Questions and Posts

Simplification of products of Dirac matrices

by: ecterrab 14540 Product: Maple

December 04 2017

4 0

The computation of traces of products of Dirac matrices was implemented years ago - see Physics,Trace .

The simplification of products of Dirac matrices, however, was not. Now it is, and illustrating this new feature is the matter of this post. To reproduce the results below please update the Physics library with the one distributed at the Maplesoft R&D Physics webpage.

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First of all, when loading Physics, a frequent question is about the signature, the default is (- - - +)

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(1)

This is important because the convention for the Algebra of Dirac Matrices depends on the signature. With the signatures (- - - +) as well as (+ - - -), the sign of the timelike component is 1

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(2)

With the signatures (+ + + -) as well as (- + + +), the sign of the timelike component is of course -1

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(3)

The simplification of products of Dirac Matrices, illustrated below with the default signature, works fine with any of these signatures, and works without having to set a representation for the Dirac matrices -- all the results are representation-independent.

The examples below, however, also illustrate a new feature of Physics, for now implemented as a Library:-PerformMatrixOperations command (there is a related, also new, command, , to just present the underlying matrix operations, without performing them). To illustrate this other new functionality , set a representation for the Dirac matrices, say the standard one

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(4)

The four Dirac matrices are

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(5)

The definition of the Dirac matrices is implemented in Maple following the conventions of Landau books ([1] Quantum Electrodynamics, V4), and does not depend on the signature, ie the form of these matrices is

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Array(%id = 18446744078360529726)

(6)

With the default signature, the space part components of change sign when compared with corresponding ones from while the timelike component remains unchanged

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(7)

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Array(%id = 18446744078677131982)

(8)

For the default signature, the algebra of the Dirac Matrices, loaded by default when Physics is loaded, is (see page 80 of [1])

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(9)

When the sign of the timelike component of the signature is -1, we have a -1 factor on the right-hand side of (9).

Note as well that in (9) the right-hand side has no matrix elements. This is standard in particle physics where the computations are performed algebraically, without performing the matrix operations. For the purpose of actually performing the underlying matrix operations, however, one may want to rewrite this algebra including a 4x4 identity matrix. For that purpose, see Algebra of Dirac Matrices with an identity matrix on the right-hand side. For the purpose of this illustration, below we proceed with the algebra as shown in (9), interpreting right-hand sides as if they involve an identity matrix.

Verify the algebra rule by performing all the involved matrix operations

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(10)

Note that, regarding the spacetime indices, this is a 4x4 matrix, whose elements are in turn 4x4 matrices. Compute first the external 4x4 matrix related to and

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Matrix(%id = 18446744078587020822)

(11)

Perform now all the matrix operations involved in each of the elements of this 4x4 matrix

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Matrix(%id = 18446744078743243942)

(12)

By eye everything checks OK.

Consider now the following five products of Dirac matrices

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(13)

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(14)

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(15)

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(16)

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(17)

New: the simplification of these products is now implemented

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(18)

Verify this result performing the underlying matrix operations

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(19)

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Matrix(%id = 18446744078553169662) = 4

(20)

The same with the other expressions

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(21)

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(22)

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Array(%id = 18446744078695012102)

(23)

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Array(%id = 18446744078701714238)

(24)

For e2

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(25)

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(26)

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Matrix(%id = 18446744078470204942)

(27)

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Matrix(%id = 18446744078550068870)

(28)

For e3 we have

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(29)

Verify this result,

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(30)

In this case, with three free spacetime indices lambda, nu, rho, the spacetime components form an array 4x4x4 of 64 components, each of which is a matrix equation

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Array(%id = 18446744078647326830)

(31)

For instance, the first element is

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(32)

and it checks OK:

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Matrix(%id = 18446744078647302614) = Matrix(%id = 18446744078647302974)

(33)

How can you test the 64 components of T all at once?

1. Compute the matrices, without displaying the whole thing, take the elements of the array and remove the indices (ie take the right-hand side); call it M

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For instance,

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Matrix(%id = 18446744078629635726) = Matrix(%id = 18446744078629636206)

(34)

Now verify all these matrix equations at once: take the elements of the arrays on each side of the equations and verify that the are the same: we expect for output just

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(35)

The same for e4

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(36)

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(37)

Regarding the spacetime indices this is now an array 4x4x4x4

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Array(%id = 18446744078730196382)

(38)

For instance the first of these 256 matrix equations

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(39)

verifies OK:

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Matrix(%id = 18446744078727227382) = Matrix(%id = 18446744078727227862)

(40)

Now all the 256 matrix equations verified at once as done for e3

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(41)

Finally, although there is more work to be done here, let's define some tensors and contract their product with these expressions involving products of Dirac matrices.

For example,

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${A[nu], B[lambda], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}$

(42)

Contract with e1 and e2 and simplify

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(43)

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(44)

Download DiracMatricesAndPerformMatrixOperation.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

a question on output of dsolve...

Asked by: mehdi jafari 774

December 04 2017

0 14

i have system of ode, which has three unkonwn function,x(t),y(t),z(t), and i need differentiation of theses function with respect to " t " , how can i use numeric solution to have these parameters? tnx for help in advance

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restart;

>	wl:=0.08:alpha:=0.5:gamma0:=8.821e10:mu0:=4Pi1e-7:gammA:=gamma0/(1+alpha^2):alphaPrime:=mu0gammAalpha: gamma1:=mu0gammA:m0:=mu0*1e5:num:=[seq](i,i=1e2..1e4):

>	w:=num[1]: dsys:={diff(x(t), t) =alphaPrime(z(t)^2wlcos(wt)/gamma1+y(t)^2wlcos(wt)/gamma1),diff(y(t),t)=-z(t)wlcos(wt)-alphaPrimey(t)wlcos(wt)x(t)/gamma1,diff(z(t),t)=y(t)wlcos(wt)-alphaPrimez(t)wlcos(wt)*x(t)/gamma1,x(0)=.01,y(0)=1,z(0)=0.1};

${diff(x(t), t) = 0.4000000000e-1*z(t)^2*cos(0.1e3*t)+0.4000000000e-1*y(t)^2*cos(0.1e3*t), diff(y(t), t) = -0.8e-1*z(t)*cos(0.1e3*t)-0.3999999999e-1*y(t)*cos(0.1e3*t)*x(t), diff(z(t), t) = 0.8e-1*y(t)*cos(0.1e3*t)-0.3999999999e-1*z(t)*cos(0.1e3*t)*x(t), x(0) = 0.1e-1, y(0) = 1, z(0) = .1}$

(1)

>

>	ans:=dsolve(dsys,numeric,output = listprocedure);

(2)

>	#i want to have d/dt(x(t),d/dt(y(t),d/dt(z(t)

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Download question.mw

Plotranges and colormap...

Asked by: Spinosaurus 120

December 04 2017

2 5

Hello everyone!

I have several problems with listcontplot.

First of all, when I plot spectrogram of my test signal I get correct result. I use colorscheme = ["#00007F", "Blue", "#007FFF", "Aqua", "#7FFF7F", "Yellow", "coral", "Red"] as colormap. But I can't understand how to set option coloring in listcontplot correctly to obtain same result, because coloring works as Gradient from ColorTools.

Secondly, I use subs to change plot ranges. My apprach consist in several steps:

plot( listdensityplot(...) );
temp := plottools:-getdata(%, 'rangesonly');
subs(temp[1] = NewX0 .. NewX1, temp[2] = NewY0 .. NewY1, listdensityplot(...) );

Is it possible to simplify this solution?

P.S. My test programm with test data.

Maple_s_test.mw

Test.txt

My Spectrograms:

Has anybody experienced Maple document freezing in...

Asked by: EmmaSkovgaard 10

December 04 2017

2 3

Hi,

With exam period coming closer and closer I am getting worried that this (see picture) might happen again with my document. The document is completed "deconstructed" all the way through, it contains it unreadable form when exported as PDF and when opened from another device.

Does anybody know what has gone wrong and what to do?

Thankyou,

Emma

Avoiding Scurvy on a Budget

by: Daniel Cheslo 140 Product: Maple

December 04 2017

3 0

Are you a broke university or college student, living day to day, sustaining yourself on a package of ramen noodles every day? Scared that you’re missing necessary nutrients? Well this is the Maple application for you!

This app, refurbished and modified to fit the budget and nutritional needs of a student, based on this older app: https://www.mapleprimes.com/maplesoftblog/34983-The-Diet-Problem, takes a list of more than 20 foods, and based on nutritional and budget constraints that you choose and can change, will give you the cheapest option for foods that fits your needs!

Maybe you’re not a broke student, but rather someone who wants to keep track of calorie intake or macronutrient intake on a daily basis, well then this app still works for you!

Find it here:
https://www.maplesoft.com/applications/view.aspx?SID=154368

How to extract data from a simple Maple plot and w...

Asked by: maltehenkel 20

December 04 2017

1 2

Suppose that I make a simple plot, of a function f(x), such as p1 := plot( 2*x+1,x=0..4,numpoints=9);

Then logically, the plot p1 contains a table with values (x, f(x)) which are plotted.

HOW can I extract that table of values from the plot p1, and write it to a data file (simple text file), so that I can process the data elsewhere (for example, for making a professional graph from them).

I had scripts who did this up to Maple 13 (or Maple V release 13, if that is the politically correct version), but they do not work anymore with more recent versions.

Many thanks !!

Malte.

Define discrete point probability distributions an...

Asked by: weidade37211 110

December 03 2017

1 3

first question:

Define a two point distribution:
f1(x):=piecewise(x = 0, 0.3, x = 0.1, 0.7, 0)

Dist1 := Distribution(PDF = f1)

R1 := RandomVariable(Dist1)

Mean(R1)

Why Mean(R1) gives 0? should be 0.07...

If I managed to define antother discrete point distribution say R2. Can I define R:=R1+R2 and then caculate Mean(R) or even plot PDF of R?

How do I take an integral with variable upper limi...

Asked by: Aaeru Michi 45

December 03 2017

1 13

I solved a system of differential equations and want to declare an integral with variable upper limit ( T(x) ) from a function I got from the system ( r(x) ). I did so, but Maple doesn't recognise it as a function of one variable, because when I put this T(x) into a system, it gives me an error:

Error, (in dsolve/numeric/process_input) input system must be an ODE system, got independent variables {t, x}

Though there is no t in the system, t was an integration variable when I declared T(x).

Where is the mistake?

2.mw

Generator of exercises with vectors

by: Lenin Araujo Castillo 1790 Product: Maple 2017

December 03 2017

3 0

Application to generate vector exercises using very easy to use patterns. These exercises are for sum of vectors in calculations of magnitude, direction, graph and projections. Basically to evaluate our students on the board and thus not lose the results. For university professors and engineering students.

Generator_of_exercises_with_vectors.mw

Lenin Araujo Castillo

Ambassador of Maple

How to convert negligible values (close to zero) i...

Asked by: Teep 200

December 03 2017

0 2

In LPSolve, I have solutions returned in matrix form that contain elements that are negligible (values close to zero).

Is there a simple way to convert and reduce these values to zero in the matrix (rather than return the exact values)?

Thanks!

How to plot in a positive integral domain and find...

Asked by: Kasuni Welihinda 5

December 03 2017

1 2

Is there a way that I can obtain a plot of a set of points like {(2,4),(3,5),(4,7),(5,8),(6,11)} in the domain of the set of positive integers. And the codomain is also the set of positive integers?

And after that is it possible to obtain a polynomial interpolation for three of the points say (2,4),(3,5) and (4,7)?

Thanks a lot in advance.

How to find the points indicated in the graph of H...

Asked by: herclau 1043

December 03 2017

0 4

Hello,
How to find the points indicated in the graph of the following function:
(1.25 * y-sqrt (abs (x))) ^ 2 + x ^ 2-1,

I have used the Taks: the Second Derivative Test. Without results, as I show below:

Heart_Critical_Points_and_the_Second_Derivative_Test.mw

Critical Points and the Second Derivative Test

Objective Function

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(1)

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List of Independent Variables

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(2)

Equations

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[-(1.25*y-abs(x)^(1/2))*abs(1, x)/abs(x)^(1/2)+2*x, 3.1250*y-2.50*abs(x)^(1/2)]

(3)

Critical Points

>	$temp := remove(has, solve([-(1.25y-abs(x)^(1/2))abs(1, x)/abs(x)^(1/2)+2x, 3.1250y-2.50*abs(x)^(1/2)], v, Explicit), I); convert({seq(eval(v, temp[k]), k = 1 .. nops(temp))}, list)$

(4)

Second Derivative Test

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(5)

Hessians and their Eigenvalues

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Temp := [Student[MultivariateCalculus][SecondDerivativeTest]((1.25*y-abs(x)^(1/2))^2+x^2-1, v = [], output = hessian)]; for k to nops(Temp) do Temp[k], convert(LinearAlgebra[Eigenvalues](Temp[k]), list) end do

Error, (in LinearAlgebra:-Eigenvalues) expecting either Matrices of rationals, rational functions, radical functions, algebraic numbers, or algebraic functions, or Matrices of complex(numeric) values

Download Heart_Critical_Points_and_the_Second_Derivative_Test.mw

Recursive Function in maple, trickery, too many le...

Asked by: Seb1123 45

December 02 2017

1 7

Hey. I can't seem to implement the following recursive (piecewise) function:

f:=n->piecewise(n=0,1,n>=1,sum(f(k),k=0..n-1))

This doesn't work..How do I make it work? :(

Help ...Maple will not plot...

Asked by: mathiaszip 15

December 02 2017

0 1

Hi

II ve managed to build an expression with one variable
I me trying to plot this expression on a defined range but maple doesn t not let me.

How do I manage to plot this expression ?

If you have any advise on how to improve my code I m open to comments

Thanks a lot in advance

>

#data

>

(1)

>

#geometry

>

#coefficients

>

#formula

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`σc` := proc (epsilon) options operator, arrow; fcm*(k*epsilon/`ϵc1`-epsilon^2/`ϵc1`^2)/(1+(k-2)*epsilon/`ϵc1`) end proc;

proc (epsilon) options operator, arrow; fcm*(k*epsilon/`ϵc1`-epsilon^2/`ϵc1`^2)/(1+(k-2)*epsilon/`ϵc1`) end proc

(2)

>

#Pressure as a function of y (y=) :

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`σc` := proc (y) options operator, arrow; fcm*(k*`ϵc`*y/(x*`ϵc1`)-`ϵc`^2*y^2/(x^2*`ϵc1`^2))/(1+(k-2)*`ϵc`*y/(x*`ϵc1`)) end proc;

proc (y) options operator, arrow; fcm*(k*`ϵc`*y/(x*`ϵc1`)-`ϵc`^2*y^2/(x^2*`ϵc1`^2))/(1+(k-2)*`ϵc`*y/(x*`ϵc1`)) end proc

(3)

>

int(25*(1302.000000*`ϵc`*x/x-226757.3696*`ϵc`^2*x^2/x^2)/(1+349.6190476*`ϵc`*x/x), x = 0 .. x)

(4)

>

C := `assuming`([int(fcm*(k*`ϵc`*s/(x*`ϵc1`)-(`ϵc`*s/(x*`ϵc1`))^2)*b/(1+(k-2)*`ϵc`*s/(x*`ϵc1`)), s = 0 .. x)], [0 < x and x <= (1/2)*d, 0 <= `ϵc` and `ϵc` <= 0.35e-2]);

-0.7487980799e-19*x*(0.2706771684e26*`ϵc`^2+0.1331955800e22*ln(874047619.*`ϵc`+2500000.)-0.1962210817e23-0.4656771182e24*`ϵc`)/`ϵc`

(5)

>

(6)

>

(7)

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(8)

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Error, (in plot) expected a range but received `ϵc` = 0 .. 0.34e-2

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Download HW1_-_EC2_strain-pressure_graph.mw

>

#data

>

(1)

>

#geometry

>

#coefficients

>

#formula

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`σc` := proc (epsilon) options operator, arrow; fcm*(k*epsilon/`ϵc1`-epsilon^2/`ϵc1`^2)/(1+(k-2)*epsilon/`ϵc1`) end proc;

proc (epsilon) options operator, arrow; fcm*(k*epsilon/`ϵc1`-epsilon^2/`ϵc1`^2)/(1+(k-2)*epsilon/`ϵc1`) end proc

(2)

>

#Pressure as a function of y (y=) :

>

`σc` := proc (y) options operator, arrow; fcm*(k*`ϵc`*y/(x*`ϵc1`)-`ϵc`^2*y^2/(x^2*`ϵc1`^2))/(1+(k-2)*`ϵc`*y/(x*`ϵc1`)) end proc;

proc (y) options operator, arrow; fcm*(k*`ϵc`*y/(x*`ϵc1`)-`ϵc`^2*y^2/(x^2*`ϵc1`^2))/(1+(k-2)*`ϵc`*y/(x*`ϵc1`)) end proc

(3)

>

int(25*(1302.000000*`ϵc`*x/x-226757.3696*`ϵc`^2*x^2/x^2)/(1+349.6190476*`ϵc`*x/x), x = 0 .. x)

(4)

>

C := `assuming`([int(fcm*(k*`ϵc`*s/(x*`ϵc1`)-(`ϵc`*s/(x*`ϵc1`))^2)*b/(1+(k-2)*`ϵc`*s/(x*`ϵc1`)), s = 0 .. x)], [0 < x and x <= (1/2)*d, 0 <= `ϵc` and `ϵc` <= 0.35e-2]);

-0.7487980799e-19*x*(0.2706771684e26*`ϵc`^2+0.1331955800e22*ln(874047619.*`ϵc`+2500000.)-0.1962210817e23-0.4656771182e24*`ϵc`)/`ϵc`

(5)

>

(6)

>

(7)

>

(8)

>

Error, (in plot) expected a range but received `ϵc` = 0 .. 0.34e-2

>

Download HW1_-_EC2_strain-pressure_graph.mw

Physics & Math with maple (DE numerically)...

Asked by: JoeyJoe 10

December 02 2017

2 21

This is my first time in this forum, so I hope I use the correct conventions. If not please notice me.

When a light body orbits a heavy body under the influence of gravity (e.g. a planet around the Sun), Newton’s laws show that the orbit is restricted to a two-dimensional plane and is given by the differential equation

d²/d(φ)²(1/r(φ)) + 1/r(φ) = GM/h²

Here, (r, φ) is the path of the light body in polar coordinates, M is the mass of the heavy body, G is the gravitational constant, and h is a constant related to the angular velocity of the light body (h = r²φ ̇). The heavy body can be considered to be approximately stationary and located at the origin.

Use Maple to solve this differential equation numerically, taking M = 1, G = 1, h = 1 with initial conditions

r(0)=2/3, r′(0)=dr/dφ (0) = 0

Using polar coordinates, create a plot of the orbit (r(φ), φ) for
0 ≤ φ < 2π. You should observe a perfect ellipse.
Since I am not a frequent maple user, I hope somebody can help me here

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