Cadabra Q&A - Recent questions and answers
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http://cadabra.science/qa/1035/matrix-exponential
<p>It would be nicer if the future Cadabra would be able to deal with calculating the exponentials of (Gamma) matrices. </p>
Feature requestshttp://cadabra.science/qa/1035/matrix-exponentialSun, 17 Feb 2019 14:18:29 +0000Examles of using Cadabra for superspace calculations.
http://cadabra.science/qa/1033/examles-of-using-cadabra-for-superspace-calculations
<p>I am trying to use a program for computation in superspace. Unfortunately, this raises many questions. I am sure that many people have already done such calculations. Can someone send examples of such calculations? For example, how the action of covariant derivatives is realized and the chiral superfield is described.</p>
General questionshttp://cadabra.science/qa/1033/examles-of-using-cadabra-for-superspace-calculationsFri, 01 Feb 2019 13:47:09 +0000Answered: Curved and Flat Indices
http://cadabra.science/qa/1031/curved-and-flat-indices?show=1032#a1032
<p>Those are just names; the full form of that first property is</p>
<pre><code> {\mu,\nu,\rho}::Indices(name=curved, position=fixed).
</code></pre>
General questionshttp://cadabra.science/qa/1031/curved-and-flat-indices?show=1032#a1032Thu, 31 Jan 2019 14:36:20 +0000Answered: Please show a simple example showing difference b/w Indices(curved) and Indices(flat).
http://cadabra.science/qa/1029/simple-example-showing-difference-indices-curved-indices?show=1030#a1030
<p>Read the chapter on indices in the reference manual,</p>
<p><a rel="nofollow" href="https://cadabra.science/notebooks/ref_indices.html">https://cadabra.science/notebooks/ref_indices.html</a></p>
<p>Hope that helps; if it doesn't, reply to this and I'll make another attempt.</p>
General questionshttp://cadabra.science/qa/1029/simple-example-showing-difference-indices-curved-indices?show=1030#a1030Tue, 29 Jan 2019 10:00:19 +0000Answered: Substitute Christoffel2nd into Dual Riemann tensor
http://cadabra.science/qa/1027/substitute-christoffel2nd-into-dual-riemann-tensor?show=1028#a1028
<p>The steps should be</p>
<pre><code>substitute(uDualRiemann, lRieman);
substitute(uDualRiemann, Riemann);
substitute(uDualRiemann,Christoffel2nd);
</code></pre>
<p>The 1st line rewrites the Riemann with all lower indices in terms of one with the first index raised. The 2nd line substitutes the definition of the Riemann in terms of Christoffel symbols. And then the 3rd line rewrites the Christoffel symbols in terms of derivatives of the metric.</p>
General questionshttp://cadabra.science/qa/1027/substitute-christoffel2nd-into-dual-riemann-tensor?show=1028#a1028Sun, 27 Jan 2019 11:41:46 +0000Answered: Fedora 29 rpm cpack error
http://cadabra.science/qa/1020/fedora-29-rpm-cpack-error?show=1023#a1023
<p>I have made a few changes to detect Fedora 29 and build the package appropriately (using normal cmake, not cmake3). The resulting package is now available from the download page.</p>
Installation troublehttp://cadabra.science/qa/1020/fedora-29-rpm-cpack-error?show=1023#a1023Mon, 14 Jan 2019 19:06:53 +0000Answered: Correlation functions, Wick theorem and summation
http://cadabra.science/qa/1012/correlation-functions-wick-theorem-and-summation?show=1018#a1018
<p>I am not sure exactly which summation problem you are trying to solve; maybe you can give a bit more detail?</p>
<p>For the Wick story, here's a bit of code I put together a while ago which may help you get things implemented. It basically gives you a <code>contract</code> function which takes a product of fields and replaces all pairs with propagators. You'll need to expand this of course to do real world problems but the gist of the solution is in here.</p>
<p>First, you need something that takes a list of numbers and returns all possible ways in <br>
which you can pair numbers:</p>
<pre><code>def all_pairs(lst):
if len(lst) < 2:
yield []
return
else:
a = lst[0]
for i in range(1,len(lst)):
pair = (a,lst[i])
for rest in all_pairs(lst[1:i]+lst[i+1:]):
yield [pair] + rest
</code></pre>
<p>That's just a bit of standard Python, nothing Cadabra-specific. Then you need something that takes a cadabra expression and does the contractions in there:</p>
<pre><code>def contract(ex):
amp:=0.
L=list(range(len(ex)))
for c in all_pairs(L):
diag:=1;
for p in c:
a1=ex[p[0]][0]
a2=ex[p[1]][0]
diag *= $G( @(a1) - @(a2) )$
amp += diag
return amp
</code></pre>
<p>Hopefully this makes it clear how to take Cadabra expressions apart and build new ones from the pieces. Then the following works:</p>
<pre><code>ex:=\phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4);
contract(ex);
</code></pre>
<p>gives</p>
<p>$$G\left(x_{1}-x_{2}\right) G\left(x_{3}-x_{4}\right)+G\left(x_{1}-x_{3}\right) G\left(x_{2}-x_{4}\right)+G\left(x_{1}-x_{4}\right) G\left(x_{2}-x_{3}\right)$$</p>
<p>For any serious work you'll probably have to rewrite the <code>all_pairs</code> function so that it is not recursive.</p>
General questionshttp://cadabra.science/qa/1012/correlation-functions-wick-theorem-and-summation?show=1018#a1018Sat, 12 Jan 2019 14:13:48 +0000Answered: Fedora 29 rpm is missing
http://cadabra.science/qa/1016/fedora-29-rpm-is-missing?show=1017#a1017
<p>I didn't yet get around to building it, will do so the next few days.</p>
Installation troublehttp://cadabra.science/qa/1016/fedora-29-rpm-is-missing?show=1017#a1017Sat, 12 Jan 2019 12:53:23 +0000Answered: whenever make Cadabra do lengthy calculations, I always get error on output
http://cadabra.science/qa/1009/whenever-cadabra-lengthy-calculations-always-error-output?show=1010#a1010
<p>You'll have to make this a bit more specific. Can you email me (info@cadabra.science) a notebook that fails in this way (preferably as short as possible)? </p>
General questionshttp://cadabra.science/qa/1009/whenever-cadabra-lengthy-calculations-always-error-output?show=1010#a1010Wed, 02 Jan 2019 17:42:33 +0000Answered: Homebrew install problem
http://cadabra.science/qa/1005/homebrew-install-problem?show=1007#a1007
<p>There is now the option to install directly using homebrew. First install MacTeX, either directly from its website, or using homebrew,</p>
<pre><code>brew tap caskroom/cask
brew cask install mactex
</code></pre>
<p>Then install cadabra using</p>
<pre><code>brew tap kpeeters/repo
brew install cadabra2
</code></pre>
<p>These instructions are now also on the download page. Please let me know if you run into any trouble; this has not yet received a lot of testing!</p>
Installation troublehttp://cadabra.science/qa/1005/homebrew-install-problem?show=1007#a1007Mon, 17 Dec 2018 20:26:02 +0000Answered: Grouping substitution rules makes the kernel unresponsive
http://cadabra.science/qa/1001/grouping-substitution-rules-makes-the-kernel-unresponsive?show=1002#a1002
<p>Those output lines suggest that you used a version which wasn't really meant for production. I am assuming you compiled from source? Can you try pulling the latest version from github, building that and then trying again?</p>
<p>If that does not solve the problem, please email your notebook to info@cadabra.science and I'll have a look. If it worked before it is probably something small.</p>
Bug reportshttp://cadabra.science/qa/1001/grouping-substitution-rules-makes-the-kernel-unresponsive?show=1002#a1002Fri, 14 Dec 2018 11:14:08 +0000Answered: collect_factors(_) for Symbol indices
http://cadabra.science/qa/998/collect_factors-_-for-symbol-indices?show=999#a999
<p>It should, it's a bug introduced with tidying up some pattern matching functions. I have just pushed a fix for this to github. Thanks for reporting.</p>
General questionshttp://cadabra.science/qa/998/collect_factors-_-for-symbol-indices?show=999#a999Mon, 03 Dec 2018 15:02:11 +0000Answered: Calculating determinants and traces
http://cadabra.science/qa/951/calculating-determinants-and-traces?show=991#a991
<p>There's a <code>Determinant</code> property now which you can use together with <code>complete</code> to compute determinants, as in <a rel="nofollow" href="https://cadabra.science/manual/Determinant.html">https://cadabra.science/manual/Determinant.html</a> .</p>
General questionshttp://cadabra.science/qa/951/calculating-determinants-and-traces?show=991#a991Fri, 23 Nov 2018 21:47:11 +0000Answered: Simple example showing differences between position=free, fixed and independent
http://cadabra.science/qa/937/example-showing-differences-between-position-independent?show=990#a990
<p>See the updated chapter 1.4 of the reference guide, <a rel="nofollow" href="https://cadabra.science/notebooks/ref_indices.html">https://cadabra.science/notebooks/ref_indices.html</a> .</p>
General questionshttp://cadabra.science/qa/937/example-showing-differences-between-position-independent?show=990#a990Fri, 23 Nov 2018 16:14:47 +0000Answered: Multiplying \sqrt{2} by 1/\sqrt{2} crashes the kernel unexpectedly
http://cadabra.science/qa/983/multiplying-sqrt-by-sqrt-2-crashes-the-kernel-unexpectedly?show=984#a984
<p>No, this is a bug. You'd be surprised how many computations can be done without ever encountering this one... Will fix, thanks.</p>
General questionshttp://cadabra.science/qa/983/multiplying-sqrt-by-sqrt-2-crashes-the-kernel-unexpectedly?show=984#a984Mon, 19 Nov 2018 21:17:52 +0000Answered: AntiCommuting/SelfAntiCommuting only for different indices
http://cadabra.science/qa/977/anticommuting-selfanticommuting-only-different-indices?show=979#a979
<p>You would first declare the $a$ and $a^\dagger$ operators to be anti-commuting among themselves, so</p>
<pre><code>{n,m,p,q}::Indices(position=free);
a_{n}::SelfAntiCommuting;
ad_{n}::SelfAntiCommuting;
ad_{n}::LaTeXForm("a^\dagger").
</code></pre>
<p>where the last line is just to make the <code>ad_{n}</code> print nicely. Then ensure that the default is for daggered and non-daggered operators to be non-commuting,</p>
<pre><code>{a_{n}, ad_{m}}::NonCommuting;
</code></pre>
<p>Now the only thing you need to do is to decide what you want to do with expressions which involve $a$ and $a^\dagger$. In many cases, the logic is to 'move' all $a$ to the far right of an expression (so they can annihilate a vacuum state on which they act). You can do that with the rule</p>
<pre><code>\delta_{m n}::KroneckerDelta;
rl:= a_{m} ad_{n} -> - ad_{n} a_{m} + \delta_{n m};
</code></pre>
<p>You then apply this rule repeatedly to an expression, distributing products at every step and eliminating Kronecker deltas. If you want you can do this automatically at every step of a calculation, by sticking that logic into the <code>post_process</code> function. Example:</p>
<pre><code>def post_process(ex):
# move all a operators to the right
converge(ex):
substitute(ex, rl)
distribute(ex)
eliminate_kronecker(ex)
sort_product(ex)
collect_terms(ex)
</code></pre>
<p>If you now enter e.g. the expression</p>
<pre><code>ex:= a_{n} ad_{p} a_{m} ad_{m} a_{n};
</code></pre>
<p>you get the answer</p>
<p>$$ -a^\dagger_m a_m a_p + \delta_{m m} a_p .$$</p>
<p>Hope this helps.</p>
General questionshttp://cadabra.science/qa/977/anticommuting-selfanticommuting-only-different-indices?show=979#a979Mon, 12 Nov 2018 20:54:18 +0000Answered: Windows 10 version doesn't work
http://cadabra.science/qa/971/windows-10-version-doesnt-work?show=972#a972
<p>This is a known issue and we're investigating the origin of it; will post here if there's something new to try. </p>
Installation troublehttp://cadabra.science/qa/971/windows-10-version-doesnt-work?show=972#a972Sun, 28 Oct 2018 20:22:24 +0000Answered: IPython notebooks example not working
http://cadabra.science/qa/963/ipython-notebooks-example-not-working?show=964#a964
<p>The <code>param</code> argument should have been optional, but at some point we stripped out the default value for that argument, and since the Cadabra preparser fills it in automatically, no-one noticed. </p>
<p>Until this is fixed, you can use</p>
<pre><code>Symmetric(ex, Ex(''))
</code></pre>
<p>in place of your last line (that is, set <code>param</code> equal to an empty expression).</p>
Bug reportshttp://cadabra.science/qa/963/ipython-notebooks-example-not-working?show=964#a964Thu, 25 Oct 2018 20:36:42 +0000Answered: drop_weight before distribute
http://cadabra.science/qa/954/drop_weight-before-distribute?show=955#a955
<p>The logic here is intentional: <code>drop_weight</code> only acts on terms with a well-defined weight, and the $\epsilon (\epsilon^3 + 5)$ term is not of that type. So at the moment the answer is 'no'.</p>
<p>However, if your problem is that doing <code>distribute</code> is going to mess up too many other terms, you can try zooming into the expression and only distributing terms which match a certain pattern. E.g.</p>
<pre><code>\epsilon::Weight(label=field, value=1);
Exp:=(A+B)*\epsilon**2+\epsilon*(\epsilon**3+5);
zoom(_, $\epsilon Q??$);
distribute(_);
unzoom(_);
drop_weight(_, $field=4$);
</code></pre>
<p>would produce $(A+B)\epsilon^2 + 5\epsilon$, with the brackets in the first term intact.</p>
General questionshttp://cadabra.science/qa/954/drop_weight-before-distribute?show=955#a955Mon, 15 Oct 2018 10:09:53 +0000Issue with combine algorithm
http://cadabra.science/qa/953/issue-with-combine-algorithm
<p>I am using cadabra 2.2.1 </p>
<p>ex:=A^\alpha B<em>{\alpha};<br>
combine(</em>);</p>
<p>produces the output (BA) which is clearly unexpected while, </p>
<p>ex:=A<em>\alpha B</em>{\alpha};<br>
combine(_);</p>
<p>produces the output (AB) which is expected. I require the combine to work for contractions like Einstein summation. Please help</p>
Bug reportshttp://cadabra.science/qa/953/issue-with-combine-algorithmSat, 13 Oct 2018 13:11:03 +0000Require two calls to canonicalise?
http://cadabra.science/qa/938/require-two-calls-to-canonicalise
<p>Hi Folks,</p>
<p>The following code requires two calls to <code>canonicalise</code> to get the expected output (zero).<br>
Is this expected behaviour? </p>
<pre><code>W^{a}_{b c}::TableauSymmetry(shape={2}, indices={1,2});
{a,b,c,d,e,f,g,h#}::Indices(position=fixed).
foo := v_{a} W^{a}_{b c} - v^{a} W_{a c b};
sort_product (foo);
canonicalise (foo);
canonicalise (foo);
</code></pre>
<p>Cheers,<br>
Leo</p>
Bug reportshttp://cadabra.science/qa/938/require-two-calls-to-canonicaliseFri, 28 Sep 2018 09:51:08 +0000Answered: Generic typesetting LaTex error on Windows 10, Cadabra2.2.1d
http://cadabra.science/qa/930/generic-typesetting-latex-error-on-windows-10-cadabra2-2-1d?show=932#a932
<p>Once that happens, can you open a command line window, navigate to the TMPDIR path, and do</p>
<pre><code>pdflatex [filename]
</code></pre>
<p>where <code>filename</code> is, in the example above, <code>sglw.2.tex</code>. Then send me the resulting error message (if any).</p>
<p>It is possible that MikTeX is installed to not download missing packages, or perhaps the <code>pdflatex</code> command is not in your path, but I can't see that from the error message you quoted.</p>
Bug reportshttp://cadabra.science/qa/930/generic-typesetting-latex-error-on-windows-10-cadabra2-2-1d?show=932#a932Thu, 20 Sep 2018 08:24:30 +0000Answered: Derivatives not inheriting SelfAntiCommuting with canonicalise
http://cadabra.science/qa/908/derivatives-inheriting-selfanticommuting-canonicalise?show=909#a909
<p>This is a bug, thanks for reporting it. The code path for <code>sort_product</code> is different from the one for <code>canonicalise</code>; the latter is more complicated and contains a bug somewhere. <br>
I have opened an issue at <a rel="nofollow" href="https://github.com/kpeeters/cadabra2/issues/113">https://github.com/kpeeters/cadabra2/issues/113</a> .</p>
Bug reportshttp://cadabra.science/qa/908/derivatives-inheriting-selfanticommuting-canonicalise?show=909#a909Fri, 24 Aug 2018 18:01:10 +0000Answered: Windows 10 64-bit install not working yet
http://cadabra.science/qa/755/windows-10-64-bit-install-not-working-yet?show=902#a902
<p>Please try the 2.2.1d installer which is now available, and let me know if that's any better.</p>
Installation troublehttp://cadabra.science/qa/755/windows-10-64-bit-install-not-working-yet?show=902#a902Mon, 20 Aug 2018 16:09:24 +0000Answered: Bug: Generic TeX error
http://cadabra.science/qa/728/bug-generic-tex-error?show=901#a901
<p>There is now a 2.2.1d installer on the download page; can anyone with issues on Windows please try that one? I think this one should finally do the trick...</p>
<p>It is probably advisable to first remove the installation directory of any previous installs.</p>
Bug reportshttp://cadabra.science/qa/728/bug-generic-tex-error?show=901#a901Mon, 20 Aug 2018 16:08:02 +0000Answered: Defining the Rarita-Schwinger action: Action outputs 0
http://cadabra.science/qa/898/defining-the-rarita-schwinger-action-action-outputs-0?show=899#a899
<p>It's mostly an issue with not putting brackets at the right places. While Cadabra uses TeX notation, you sometimes have to be a little more explicit than strictly necessary in TeX, in order to avoid ambiguity. So you have to write e.g.</p>
<pre><code>\bar{\psi_{\mu}}
</code></pre>
<p>so that it is clear to Cadabra that the <code>\psi_{\mu}</code> is an 'argument' of <code>\bar</code>. Ditto for <code>\partial</code>, which you need to write as</p>
<pre><code>\partial_{\nu}{ \psi_{\rho} }
</code></pre>
<p>so that the <code>\psi_{\rho}</code> explicitly becomes an 'argument' of <code>\partial</code>. In your example, the <code>\partial_{\nu}\psi_{\rho}</code> was interpreted as a derivative acting on nothing, multiplied with <code>\psi_{\rho}</code>, giving nothing.</p>
<p>Here's a tidied-up version which works:</p>
<pre><code>#Rarita-Scwhinger Action
def post_process(ex):
sort_product(ex)
eliminate_kronecker(ex)
canonicalise(ex)
collect_terms(ex)
{\mu,\nu,\rho}::Indices(position=independent);
x::Coordinate;
\Gamma{#}::GammaMatrix(metric=\eta);
\eta_{\mu\nu}::Metric;
\partial{#}::Derivative;
\psi_{\mu}::Spinor;
\bar{#}::DiracBar;
\psi_{\mu}::Depends(x);
S:=\int{\bar{\psi_{\mu}} \Gamma^{\mu\nu\rho} \partial_{\nu}{\psi_{\rho}} }{x};
</code></pre>
<p>I have fixed a few typos and also changed the index type to 'independent' to prevent Cadabra from raising/lowering index pairs (usually better for susy computations, as you can then keep all indices on the Gamma's upstairs and more easily convert them using vielbeine).</p>
General questionshttp://cadabra.science/qa/898/defining-the-rarita-schwinger-action-action-outputs-0?show=899#a899Mon, 20 Aug 2018 10:03:08 +0000Answered: Partial derivative of exp()
http://cadabra.science/qa/895/partial-derivative-of-exp?show=896#a896
<p>This is something somewhere in the middle between an abstract and a component computation, and I haven't yet settled completely on what is the best way to handle this. For the time being I would suggest you use a simple substitution rule, e.g.</p>
<pre><code>\partial{#}::PartialDerivative;
test:=\partial_{\mu}{\exp(i*x^{\lambda}*k_{\lambda})};
rl:= \partial_{\mu}{ \exp( A?? ) } -> \exp( A?? ) \partial_{\mu}{ A?? };
substitute(test, rl);
product_rule(test);
substitute(test, $\partial_{\mu}{x^{\lambda}} -> \delta_{\mu}^{\lambda}$ );
unwrap(test);
</code></pre>
<p>That gets you almost there, but <code>eliminate_kronecker</code> which you would now want to use cannot handle the duplicate dummy index pair. So you'll need another substitution to get rid of the Kronecker delta.</p>
<p>Good example though, will see if we can make this work more easily.</p>
General questionshttp://cadabra.science/qa/895/partial-derivative-of-exp?show=896#a896Fri, 17 Aug 2018 09:22:45 +0000Answered: Is number_of_terms deprecated?
http://cadabra.science/qa/892/is-number_of_terms-deprecated?show=893#a893
<p>Yes, use standard Python <code>len</code>:</p>
<pre><code>ex:=A+B+C;
len(ex);
</code></pre>
<p>shows '3'.</p>
General questionshttp://cadabra.science/qa/892/is-number_of_terms-deprecated?show=893#a893Thu, 16 Aug 2018 07:03:49 +0000Answered: How to evaluate components of covariant derivative
http://cadabra.science/qa/885/how-to-evaluate-components-of-covariant-derivative?show=887#a887
<p>Yes you will need to determine the components of the metric and Christoffel symbols first, following e.g. the Schwarzschild tutorial notebook. You then need to expand your covariant derivatives into partial derivatives and Christoffels. You can do that with a simple substitution rule if you only have vectors and co-vectors, or you can use a more generic routine to do arbitrary tensors; see the example at the bottom of</p>
<p><a rel="nofollow" href="https://cadabra.science/notebooks/ref_programming.html">https://cadabra.science/notebooks/ref_programming.html</a></p>
<p>for more on that. This is something that will become available in the form of a package in 2.2.2 or 2.2.4.</p>
General questionshttp://cadabra.science/qa/885/how-to-evaluate-components-of-covariant-derivative?show=887#a887Sun, 12 Aug 2018 09:00:34 +0000Answered: Operator acting on tensors
http://cadabra.science/qa/879/operator-acting-on-tensors?show=880#a880
<p>Adding indices to object wildcards is not supported because there are very few cases where that actually makes much sense. The problem is that, if your rule had worked, it would also match</p>
<pre><code> met{ A_{b} }{ B_{b} }
met{ A_{b c d} }{ B_{c} }
</code></pre>
<p>and so on. You then almost always want to have control over where the extra $a$ index would need to be added (at the beginning of the set? at the end? in the middle?) and the notation which you wanted to use does not leave any way to specify that location. </p>
<p>So instead, you will need to make a rule which does not take an arbitrary object, but something which has the index structure spelled out. If you had in mind just to act on symbols $V$ and $W$, the rule</p>
<pre><code>Expmet:= met{X?}{Y?} -> X?_{a} Y?_{a};
</code></pre>
<p>would work. This will apply to your <code>met{V}{W}</code> case, but will not apply to any of the cases at the top of this reply. You could, however, write e.g.</p>
<pre><code>Expmet2:= met{ X?_{b} }{ Y?_{b} } -> X?_{b a} Y?_{a b};
</code></pre>
<p>or similar. That rule would allow you to do</p>
<pre><code>{a,b,c,d}::Indices;
ex:= met{ A_{c} }{ B_{c} };
exmet:= met{ X?_{b} }{ Y?_{b} } -> X?_{b a} Y?_{a b} ;
substitute(ex, exmet);
</code></pre>
<p>Hope the logic is clear.</p>
<p>So <strong>TL;DR</strong>: object patterns (patterns with '<code>??</code>' attached) are very brute force hammers, not always suitable when name patterns (patterns with just '<code>?</code>') will do.</p>
<p>(In addition, your example triggered a parsing bug that led to the indices being put on the product, not on the individual tensors, but that's a different story).</p>
General questionshttp://cadabra.science/qa/879/operator-acting-on-tensors?show=880#a880Sat, 11 Aug 2018 10:15:16 +0000Answered: How to evaluate the tensor of energy momentum for provided metric anzatz and 4-velocity
http://cadabra.science/qa/873/evaluate-energy-momentum-provided-metric-anzatz-velocity?show=874#a874
<p>Combine the rules for <code>u</code> and <code>g</code> into one call to <code>evaluate</code>, as in</p>
<pre><code>evaluate(T, u+g, rhsonly=True);
</code></pre>
General questionshttp://cadabra.science/qa/873/evaluate-energy-momentum-provided-metric-anzatz-velocity?show=874#a874Thu, 09 Aug 2018 16:34:53 +0000Answered: Code for general Einstein field equations with diagonal metric anzatc
http://cadabra.science/qa/870/general-einstein-field-equations-with-diagonal-metric-anzatc?show=871#a871
<p>Thanks for reporting this. The main issue is a bug in computing dependencies of expressions like <code>exp(F(r))</code>, which fail to figure out that this depends on <code>r</code>. I have just pushed a fix to github which solves this (your notebook then reproduces the Ricci tensor/scalar results in that PDF file). </p>
<p>The last error (when computing the Einstein tensor) is something weird in the bridge to sympy. Will look into that.</p>
General questionshttp://cadabra.science/qa/870/general-einstein-field-equations-with-diagonal-metric-anzatc?show=871#a871Wed, 08 Aug 2018 10:25:13 +0000Answered: Unwrap applied on expresion with partial derivative and two single variable functions
http://cadabra.science/qa/854/applied-expresion-partial-derivative-variable-functions?show=855#a855
<p>It's a bug :-( That is to say, your code should have read</p>
<pre><code>{x,y}::Coordinate.
\partial{#}::PartialDerivative.
f::Depends(x).
g::Depends(y).
ex:=\partial_{x}{A*f*g};
unwrap(_);
ex:=\partial_{y}{A*f*g};
unwrap(_);
</code></pre>
<p>(changed the two <code>Depends</code> lines) but even then it would produce the wrong answer.</p>
<p>This got introduced in the 1.x -> 2.x rewrite and somehow escaped the tests. I have just pushed a fix to github now which fixes this. </p>
General questionshttp://cadabra.science/qa/854/applied-expresion-partial-derivative-variable-functions?show=855#a855Mon, 06 Aug 2018 10:40:54 +0000Answered: how can i tell cadabra that the partial derivatives commute upto linear order while calculating Riemann Tensor
http://cadabra.science/qa/850/cadabra-partial-derivatives-commute-calculating-riemann?show=851#a851
<p>That one had me puzzled for a minute... The solution is to match your brackets: both terms should have an additional <code>}</code> to close the argument of the outmost derivative. So</p>
<pre><code>\partial{#}::PartialDerivative;
c1:= \partial_{\alpha}{\partial_{\beta}{h_{\mu \nu}}} =
\partial_{\beta}{\partial_{\alpha}{h_{\mu \nu}}};
</code></pre>
<p>And to answer the question in the subject: you can then make these terms equal with a call to <code>canonicalise(c1)</code>.</p>
General questionshttp://cadabra.science/qa/850/cadabra-partial-derivatives-commute-calculating-riemann?show=851#a851Wed, 01 Aug 2018 19:32:20 +0000Answered: eliminate_kronecker(_) crashes on double derivatives
http://cadabra.science/qa/841/eliminate_kronecker-_-crashes-on-double-derivatives?show=842#a842
<p>When I run this, it correctly tells me that there is a repeated upper or lower index of fixed type (so there is an error message, but not a crash). If you want this to work with the index positions as they stand, you need to declare the index type as <code>position=free</code>. Fixed type indices of a given name can only occur once in super or subscript position.</p>
Bug reportshttp://cadabra.science/qa/841/eliminate_kronecker-_-crashes-on-double-derivatives?show=842#a842Tue, 31 Jul 2018 09:10:42 +0000Answered: noncomuting objects with derivatives are not conmuting
http://cadabra.science/qa/838/noncomuting-objects-with-derivatives-are-not-conmuting?show=839#a839
<p>Yes, that's a bug, or rather an incompleteness, as I haven't had time to sort out what is the right thing to do for inheritance of non-commutativity that works in all situations. The problem is that the partial derivative does not automatically inherit the commutativity properties of its arguments. </p>
<p>For the time being, you can add</p>
<pre><code>\partial{#}::SelfNonCommuting;
</code></pre>
<p>which enforces that any partial derivative does not commute with any other, or do something more restrictive, e.g.</p>
<pre><code>\partial_{J}{A{#}}::SelfNonCommuting;
</code></pre>
Bug reportshttp://cadabra.science/qa/838/noncomuting-objects-with-derivatives-are-not-conmuting?show=839#a839Thu, 26 Jul 2018 20:48:53 +0000Answered: Exponential series (or string type to expression type)
http://cadabra.science/qa/835/exponential-series-or-string-type-to-expression-type?show=836#a836
<p>That's easy: you can create a Cadabra expression object from a string by creating an <code>Ex</code> object with the string as argument. So make your last line read</p>
<pre><code>return Ex(expOutput)
</code></pre>
<p>and the result will be a Cadabra expression. </p>
<p>Should probably go into the reference guide, thanks for reminding me ;-)</p>
General questionshttp://cadabra.science/qa/835/exponential-series-or-string-type-to-expression-type?show=836#a836Tue, 24 Jul 2018 17:22:10 +0000Answered: Edit the output
http://cadabra.science/qa/823/edit-the-output?show=824#a824
<p>No, you can't, but that's a deliberate decision, as cutting and pasting makes your<br>
notebook very unreproducible (you cannot change something at the top and have the<br>
effects of that change automatically propagate to the remainder). But there is a better<br>
(IMHO) way to do something with a similar effect. I have started writing something more about this a chapter of the new reference manual, see</p>
<p><a rel="nofollow" href="https://cadabra.science/notebooks/ref_selecting.html">https://cadabra.science/notebooks/ref_selecting.html</a></p>
General questionshttp://cadabra.science/qa/823/edit-the-output?show=824#a824Sat, 14 Jul 2018 16:22:30 +0000Answered: Substitute and dummy indices manipulation
http://cadabra.science/qa/803/substitute-and-dummy-indices-manipulation?show=815#a815
<p>I can't reproduce that problem here with the version currently in github. The substitution works fine. </p>
Bug reportshttp://cadabra.science/qa/803/substitute-and-dummy-indices-manipulation?show=815#a815Tue, 10 Jul 2018 20:20:05 +0000Answered: Variable can be handled like values?
http://cadabra.science/qa/813/variable-can-be-handled-like-values?show=814#a814
<p>Yes, just make sure that you declare <code>z</code> to be a Coordinate, so add as your first line</p>
<pre><code>z::Coordinate;
</code></pre>
<p>and then all works as you expected.</p>
General questionshttp://cadabra.science/qa/813/variable-can-be-handled-like-values?show=814#a814Tue, 10 Jul 2018 20:18:09 +0000Answered: Gtk interface doesn't work.
http://cadabra.science/qa/804/gtk-interface-doesnt-work?show=812#a812
<p>I have pushed changes to github which (at least on my mac) solve this problem. Let me know if they fix it for you too.</p>
Installation troublehttp://cadabra.science/qa/804/gtk-interface-doesnt-work?show=812#a812Tue, 10 Jul 2018 15:51:47 +0000Answered: How can I implement cyclicity of trace?
http://cadabra.science/qa/729/how-can-i-implement-cyclicity-of-trace?show=802#a802
<p>Not perfect... but helpful.</p>
<p>A time ago I was trying something related to your question. After a while of looking on the web (and asking... just like you!) I came out with the following: define a circular function.</p>
<p>Let $a$ and $b$ be noncommuting, but distributable</p>
<pre><code>{a,b}::NonCommuting.
{a,b}::Distributable.
def post_process(ex):
expand_power(ex)
distribute(ex)
</code></pre>
<p>I define the function</p>
<pre><code>def isCircular(arr1, arr2):
return arr1 in arr2+' '+arr2
</code></pre>
<p>and a expression to test</p>
<pre><code>expr := (a + b)**2;
</code></pre>
<p>The trick is now to convert this to a string, split it, and compare term by term if they are circular. <strong>NOTE:</strong> blank spaces before and after the plus sign inside the <code>split</code> method.</p>
<pre><code>list = str(expr)
list = str(expr).split(' + ')
list;
for i in range(len(list)):
for j in range(i+1,len(list)):
if isCircular( list[i], list[j]):
list[j] = list[i]
list;
</code></pre>
<p>Then, convert it back to a <em>cadabra</em> expression to finally simplify. <strong>NOTE:</strong> the <code>newexpr</code> is initalised as an empty expression, it is <em>not</em> a single double quote, but double single quotes. </p>
<pre><code>newexpr = '';
for word in list:
newexpr = newexpr + ' + ' + word
;
cab_expr = Ex(newexpr)
collect_terms(cab_expr);
</code></pre>
<p>I tried something like <code>expr := (a b + b a)**2;</code> and the result is almost there! It could be a starting point!!!</p>
<p>Cheers.</p>
General questionshttp://cadabra.science/qa/729/how-can-i-implement-cyclicity-of-trace?show=802#a802Wed, 04 Jul 2018 09:51:43 +0000Answered: Gamma_5 matrix
http://cadabra.science/qa/783/gamma_5-matrix?show=800#a800
<p>If you're working on four dimensions, $\gamma^5$ is proportional to the product of the four (elemental) gamma's. Therefore you can define something like</p>
<pre><code>{s,r,l,k,m,n,p}::Indices(vector);
{s,r,l,k,m,n,p}::Integer(0..3);
\Gamma_{#}::GammaMatrix(metric=\delta);
e_{\mu\nu\lambda\rho}::EpsilonTensor(delta=\delta);
\delta_{m n}::KroneckerDelta;
g5 := e_{l m n p} \Gamma^{l m n p};
</code></pre>
<p>You have to normalise this definition accordingly. </p>
<p>Remember that <em>Cadabra</em> has a definition for the imaginary number</p>
<pre><code>i::ImaginaryI;
</code></pre>
General questionshttp://cadabra.science/qa/783/gamma_5-matrix?show=800#a800Tue, 03 Jul 2018 15:30:11 +0000Answered: Simple commutator in quantum mechanics
http://cadabra.science/qa/798/simple-commutator-in-quantum-mechanics?show=799#a799
<p>The <code>simplify</code> algorithm invokes Sympy, but does not (yet) translate all Cadabra properties to Sympy ones (Cadabra's properties are more general, so the map isn't even always possible, but of course in this particular case that's not the problem). For the time being, avoid <code>simplify</code>, as it will give wrong answers with non-commuting objects.</p>
<p>To do computations with non-commuting objects, you may want to have a look at the tutorial on the Poincare algebra, <a rel="nofollow" href="https://cadabra.science/notebooks/poincare_algebra.html">https://cadabra.science/notebooks/poincare_algebra.html</a> . That's a bit more advanced than what you asked for, but perhaps it gets you in the right direction. If not, please ask again and tell in a bit more detail what you want to compute.</p>
General questionshttp://cadabra.science/qa/798/simple-commutator-in-quantum-mechanics?show=799#a799Mon, 02 Jul 2018 23:03:20 +0000Answered: odd behaviour with indexbracket
http://cadabra.science/qa/781/odd-behaviour-with-indexbracket?show=782#a782
<p>You may want to upgrade to 2.x, where this works fine. If you can't do that, can you explain what is stopping you? (e.g. some feature which is broken/missing in 2.x but working in 1.x).</p>
Bug reportshttp://cadabra.science/qa/781/odd-behaviour-with-indexbracket?show=782#a782Tue, 12 Jun 2018 09:26:34 +0000Answered: Cadabra 2 syntax for "<<"
http://cadabra.science/qa/776/cadabra-2-syntax-for?show=778#a778
<p>Dominic has figured out how to do this with the standard Python import functionality. I have just pushed some experimental code to github which does this. With this you can import notebooks into other notebooks. For example, if you have a <code>library.cnb</code> notebook containing a cell</p>
<p><code>library.cnb</code>:</p>
<pre><code>{m, n}::Indices;
ex:=A_{m n};
</code></pre>
<p>you can import this expression and the property declaration into another notebook doing</p>
<pre><code>from library import *
</code></pre>
<p>This will auto-convert your <code>library.cnb</code> into a <code>library.py</code> file which is then imported.</p>
<p>As I said: experimental. Any feedback welcome.</p>
General questionshttp://cadabra.science/qa/776/cadabra-2-syntax-for?show=778#a778Mon, 11 Jun 2018 18:27:07 +0000Answered: Custom symbol name as index
http://cadabra.science/qa/774/custom-symbol-name-as-index?show=775#a775
<p>Ouch, that was a <em>very</em> trivial typo in the 2.x code, but sufficient to mess up all LaTeXForm'ed symbols in sub/superscripts. Now fixed in github. Thanks!</p>
General questionshttp://cadabra.science/qa/774/custom-symbol-name-as-index?show=775#a775Sun, 10 Jun 2018 09:49:17 +0000Answered: Bug in \sqrt in cadabra v1.x
http://cadabra.science/qa/769/bug-in-sqrt-in-cadabra-v1-x?show=770#a770
<p>Thanks. This works fine in 2.x, indeed seems to be a display problem. I can't tell you if and when I have time to fix this for 1.x. Perhaps time to upgrade?</p>
Bug reportshttp://cadabra.science/qa/769/bug-in-sqrt-in-cadabra-v1-x?show=770#a770Thu, 07 Jun 2018 06:01:00 +0000Answered: Bug in @unwrap for Cadabra 1.42
http://cadabra.science/qa/761/bug-in-%40unwrap-for-cadabra-1-42?show=762#a762
<p>Thanks, it's actually a bug in 2.x too... Will fix.</p>
Bug reportshttp://cadabra.science/qa/761/bug-in-%40unwrap-for-cadabra-1-42?show=762#a762Tue, 05 Jun 2018 16:26:27 +0000Answered: "g" not registered.
http://cadabra.science/qa/758/g-not-registered?show=759#a759
<p>Can you send me (info@cadabra.science) a minimal notebook which exhibits the problem?</p>
General questionshttp://cadabra.science/qa/758/g-not-registered?show=759#a759Tue, 05 Jun 2018 13:22:37 +0000