@hmc54 
Sometimes there is a wrong sign in the lecture notes or even in a textbook. The so-called Standard representation of the Dirac matrices is universal - or at least I am not aware of any different definition for them (if you have a textbook that defines the Standard representation differently, maybe you can put the reference here?). The expression in terms of space indices depends, naturally, on the signature. Still, Maple takes your choice of signature into account, and for any (Minkowski) signature, the signs of the Pauli matrices are opposite.

So I suggest you check the signature (is it (- - - +) or (+ + + -)), but more importantly: check the explicit form of the Dirac matrices when they are shown as 4 x 4, not 2 x 2 matrices. That explicit form of the contravariant Dirac matrices is shown as equation (5) on the help page for Dgamma, and for the signature used, (- - - +), if you check the covariant ones, the three space ones change sign.

You will see the same explicit form on the Wikipedia page for the Dirac matrices.

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

@hmc54 

You say, "The tetrads package you mention looks like it might offer just what I need, I'll have a read." Indeed, see Misner, Thorne and Wheeler, page 286, section 11.6 on Riemann Normal Coordinates. The Physics:-Tetrads is an implementation of the locally inertial system presented there; plus related computational tools (e.g. the Ricci coefficients).

But watching the image you attached, the missing key is to tell how you compute the components of a spacetime tensor in the tetrads (locally inertial) system. Without that explicit specification relating the components in an arbitrary spacetime referential and the tetrad one, you cannot move forward.

The implementation of Physics:-Tetrads follows paragraph 98 of Landau's book. There you read that you get the components of a tensor in one system from the ones in the other one just multiplying by the tetrad. Good. The Maple implementation follows that as shown in the first answer. Furthermore, it suffices for you to use tetrad indices to give that mathematical meaning to an otherwise generic spacetime tensor. So A[mu] represents e_[~a, mu] A[a] (has those components, e.g. for TensorArray) and A[a] has for components those of e_[a, ~mu] A[mu], and Riemann[a, b, c, d] is the object you are talking about, represents the components of Riemann in the locally inertial system.

But there is a CAVEAT: all the implementation makes sense provided A is a tensor, which of course is not the case of the Christoffel symbols, so Christoffel[a,b,c] is an object tricky to define: its definition is not just three tetrad factors multiplying Christoffel[mu, alpha, beta], but the following

where you see, on the LHS, Christoffel with tetrad indices a,b,c, representing its components in the tetrad system, and on the right-hand side, you see Christoffel with indices alpha, mu, nu, representing it in an arbitrary spacetime referential, multiplied by the expected three tetrad factors, but there are also two other terms (one of them involving the Ricci coefficients) accounting for the fact that Christoffel is not a tensor. You also cannot contract the tetrad factors with the spacetime indices of Christoffel[~alpha,mu,nu] because as said is not a tensor.

With this definition above, the formula for the covariant derivative using tetrad indices (so in the tetrad system) is the same one we have when using spacetime indices, and all the components of Christoffel[a,b,c] are equal to zero (so D_[a] = d_[a]), but the first derivatives of Christoffel[a,b,c are not all zero.

I suggest you look at the help page for the Ricci coefficients (?Tetrads:-gamma_) and follow its examples, where I show how to derive identities for the Riemann tensor in this locally inertial system.

Besides that, I will probably revise and prepare a worksheet and post it here in Mapleprimes showing how you derive the formula above for Christoffel[a,b,c] in the tetrad system. I have not seen it in textbooks, or on the web after a quick search. There may exist a simpler formula; please post it if you are aware of it.

PS: by the way there is a typographical mistake in the first formula you present from a textbook: in the second term, the last index in Christoffel is not nu but sigma.

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

@burgarth 

I couldn't download the PhysRevLett paper you mentioned. Would you mind please sending it via email or posting it here? Regarding the question, after fixing the mentioned weakness in the code, I wanted to point out that what you call the boundary terms (double sum with a KroneckerDelta inside) cannot be simplified because that would be wrong, resulting in an additional term that should be equal to 0.

In advance to taking a look at the PhysRevLett paper, I have an idea, mainly around splitting the sum from 1 to d into a sum from 1 to d-1 plus the d term, which in this case is equal to 0. The output would be as you see in my answer above after doing that replacement manually. That would be a good improvement, independent of your example, and would always be mathematically correct.

Now on your questions:

1) Yes, you can tell Maple about the dimension of the related Hilbert space. See the help page ?Physics,Setup, the section on quantumbasisdimension. But the simplifier of sums is not looking there at this moment. From this conversation, it would be good to make the simplifier look there, independent of the solution I mentioned in the previous paragraph.

2) To tell Maple that c[0] = 0, ... the simplest way is to assign it, as in c[0] := 0, c[d+1] = 0, then in the sum defining H use c[n], but again the Simplifier is not looking for 0 at the top of the sum. To indicate they are real, Setup realobjects is correct; also Assume(c[0], real) is also correct. You can always check whether the system understood your assumption with the getassumptions command, e.g. getassumptions(c[0]) after Assume(c[0],real); by the way in the context of Assume, c[0]::real means the same as C[0],real.

I will try the solution I have in mind (second paragraph) and write here again today or tomorrow.

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

@Carl Love 
I see. It seems I read too quickly his post.

Hi @Carl Love 
I see you are answering me. What I said is not that "a moderator deleted" anything, only that "I don't know if this question was accidentally deleted, but I don't see it anymore"; then that "Maybe @nm could say something ?"

By the way, I don't recall @nm deleting one of his own undisputable bug reports. Much less within only 24 hours of posting it. I also have in high regard @nm's questions, and he is also well aware that bugs he reports are always fixed, quickly, as it happened with this one. I don't know what happened here.

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

I'm curious about this too. Maybe @nm could say something? I see that his comments in the Unboxing Maple 2022 post are also absent at this moment.

One of the most important mathematical formulations in human history is that of the Standard Model in particle physics. It describes all the elementary particles (leptons like the electron, quarks, bosons as the Higgs or the photon), which in different arrangements, form all the observable particles in nature. The formulation is not just a tremendous theoretical achievement that rendered Nobel prizes but also a practical one. Basically, all the measurements performed in the particle accelerators at CERN and the Fermilab take this mathematical, abstract formulation as the starting point. However, for computer algebra systems, the complexity of the model is somewhat extreme: is not only the number of terms in the corresponding Lagrangian impressively large but also the mathematical properties of each of these objects that represented an insurmountable challenge for a long time. With hacks of different kinds, the representation of only portions of the Standard Model was possible with minimal computational capabilities.

Hidden among the novelties of Maple 2022, a breakthrough in computer algebra is the introduction, for the first time, of a representation for the whole Standard Model. This representation is fully computable, including the accessory commands to calculate related scattering amplitudes  (the essence of the computations behind particle collision experiments) and related Feynman integrals. This is a remarkable achievement in computational physics. And from the educational point of view, it brings one more brick of knowledge from "the dark side" of the moon into"the bright side." Making the Standard Model computations be at the tip of one's fingers completely transforms the possible experience we can have with the underlying knowledge.

This new development is illustrated in the Mapleprimes post The Standard Model of Particle Physics in Maple 2022.

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

Download casesplit.mw

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

@nm 
No, it is not expected that simplifying/size checks the syntax of the functions entering an expression; as explained in its help page, it only does a structural analysis of it to shorten its length. Thus, simplify(eq, size) will return the shortening it achieves without any error message. In contrast, simplify(eq) will explicitly call the routine to simplify integrals among others, and the syntax will get checked at some point, resulting in the error message you see.

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

@Kevin Dragnet 

Indeed there is something wrong with the help. It is now tracked in the database of issues, to appear fixed in the upcoming release. Meantime you can access the page directly, either entering ?Mini-Course, or from the Physics help browser colum that appears on the left, as per this image:

I will give a look at the question you mentioned about Alias later today.

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

Hi
Entering ?Mini-Course I get the right help page (the mini-course). This page is also linked in all the Physics commands' help pages. It might be that one of them has a wrong hyperlink ... Exactly how are you trying to enter this page?

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

@lemelinm 

It actually appears exactly as you show. Use the Vectors:-Component for the lhs and the LeviCivita and d_ for the rhs.

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

@Carl Love 
:):):) !!! I sometimes have these gaffes! Indeed the documentation of this package is good, I think but indeed this help page for ChangeCoordinates is just missing! 

The command was introduced in Maple 2015, and it is mentioned there in the page ?updates,Maple2015,Physics, in the section for the Vectors package, also with examples of its use - the basics for its help page. I will move that material into a new help page filling the gap.

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

Hi, would you mind please uploading a Maple worksheet with your input/output/expectation? Also, explain please what you do mean by "we want Maple to reconstruct them" Reconstruct what? H? e? What is the meaning of zeta[2] [e 0 , e 1 ]? 

To upload your worksheet please use the Green arrow. Uploading your worksheet helps to help you, avoiding others having to retype or copy and paste things that may not be correct Maple syntax.

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

@vv 

Indeed, I agree with @Rouben Rostamian, yours is beautiful and inspiring tackle!

Best!

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