1470 views

Skip to first unread message

Jan 29, 2011, 6:24:47 AM1/29/11

to xAct Tensor Computer Algebra

Hi I use xAct to calculate scattering amplitudes and as such need to

make replacements of contractions of tensors e.g.

rule= MakeRule[{p[a] p[-a], }, MetricOn -> All, ContractMetrics ->

True];

I have been experiencing issues with this feature and have

experimented extensively with it so that I could one day ask for help

properly here.

The main problem seem to be that xAct will only do this replacement if

the two tensors that are contracted are contiguous i.e.

p[a] q[c] p[-a] often wont get replaced straight away and I need to do

several rounds of Expand and Simplification. To help this process and

for clarity I often replace the inner products by parameters

(mandelstam variables for those who know about them).

However I was wondering if there might be a better way, I have two

ideas but no idea how to implement them:

1. Inserting some kind of wildcard in the definition: MakeRule[{p[a]

WILDCARD p[-a], }, MetricOn -> All, ContractMetrics -> True];

2. Running a routine that can permute the order in which

Simplification considers expressions. I believe this can be done as

Mathematica has ways of selecting terms of an expression, but have no

idea of how to do it efficiently (perhaps with a random number

generator).

Any ideas or is there already an implemented way to do this?

make replacements of contractions of tensors e.g.

rule= MakeRule[{p[a] p[-a], }, MetricOn -> All, ContractMetrics ->

True];

I have been experiencing issues with this feature and have

experimented extensively with it so that I could one day ask for help

properly here.

The main problem seem to be that xAct will only do this replacement if

the two tensors that are contracted are contiguous i.e.

p[a] q[c] p[-a] often wont get replaced straight away and I need to do

several rounds of Expand and Simplification. To help this process and

for clarity I often replace the inner products by parameters

(mandelstam variables for those who know about them).

However I was wondering if there might be a better way, I have two

ideas but no idea how to implement them:

1. Inserting some kind of wildcard in the definition: MakeRule[{p[a]

WILDCARD p[-a], }, MetricOn -> All, ContractMetrics -> True];

2. Running a routine that can permute the order in which

Simplification considers expressions. I believe this can be done as

Mathematica has ways of selecting terms of an expression, but have no

idea of how to do it efficiently (perhaps with a random number

generator).

Any ideas or is there already an implemented way to do this?

Jan 29, 2011, 1:10:41 PM1/29/11

to xAct Tensor Computer Algebra

Thanks for reporting this. Without a particular example I don´t really

know what might be the problem you find.

MakeRule is just a constructor of standard Mathematica rules, designed

to simplify or automate some simple and frequent cases of tensorial

rules. But often MakeRule is not flexible enough to implement what you

want, and then you have to write the rules yourself, using the full

power of the pattern matching capabilities of Mathematica. I believe

Mathematica's language allows you to implement any rule you might

want, but certainly not MakeRule.

In your case, the possible reorderings of your momenta are controlled

by the attributes Flat and Orderless of Times, and hence it should be

automatic. If it doesn't work for you, it might be because something

else in the rule is interfering with that, but I don't know what it

might be. As I said, the best thing would be to show an example.

As for the wildcar, I guess you mean a pattern like rest_. (note the

dot), or if you want a pattern for momenta, something like mom_?

xTensorQ[ind_] , which accepts any tensor with only one index. You can

also add tests in the index to select indices of various types.

Cheers,

Jose.

know what might be the problem you find.

MakeRule is just a constructor of standard Mathematica rules, designed

to simplify or automate some simple and frequent cases of tensorial

rules. But often MakeRule is not flexible enough to implement what you

want, and then you have to write the rules yourself, using the full

power of the pattern matching capabilities of Mathematica. I believe

Mathematica's language allows you to implement any rule you might

want, but certainly not MakeRule.

In your case, the possible reorderings of your momenta are controlled

by the attributes Flat and Orderless of Times, and hence it should be

automatic. If it doesn't work for you, it might be because something

else in the rule is interfering with that, but I don't know what it

might be. As I said, the best thing would be to show an example.

As for the wildcar, I guess you mean a pattern like rest_. (note the

dot), or if you want a pattern for momenta, something like mom_?

xTensorQ[ind_] , which accepts any tensor with only one index. You can

also add tests in the index to select indices of various types.

Cheers,

Jose.

Feb 5, 2011, 4:19:43 PM2/5/11

to xAct Tensor Computer Algebra

Thanks so much for the reply, I admit I am not fully familiar with the

Mathematica syntax due to lack of time.

So for instance an example of this problem is when I have contractions

in the denominator with the scalar head.

Something like this will do the trick:

Expand[Simplification[ContractMetric[dg1, \[Eta]] //. onshell] //.

onshell //. mand] //. onshell //. mand // Simplification

But soething like this: Simplification[

Expand[ContractMetric[dg1, \[Eta]]] //. onshell //. mand]

will leave the contractions in the denominator unreplaced.

onshell and mand are lists of rules compiled with FoldedRule.

On a different note, what exactly is SeparateMetric? Is it the

opposite of ContractMetric? I don't understand its syntax and I get

funny indices with numbers when I try it.

Thanks again

Mathematica syntax due to lack of time.

So for instance an example of this problem is when I have contractions

in the denominator with the scalar head.

Something like this will do the trick:

Expand[Simplification[ContractMetric[dg1, \[Eta]] //. onshell] //.

onshell //. mand] //. onshell //. mand // Simplification

But soething like this: Simplification[

Expand[ContractMetric[dg1, \[Eta]]] //. onshell //. mand]

will leave the contractions in the denominator unreplaced.

onshell and mand are lists of rules compiled with FoldedRule.

On a different note, what exactly is SeparateMetric? Is it the

opposite of ContractMetric? I don't understand its syntax and I get

funny indices with numbers when I try it.

Thanks again

Feb 5, 2011, 5:08:21 PM2/5/11

to Edu Serna, xAct Tensor Computer Algebra

Dealing with scalar contractions in the denominator is tricky. The problem is that Mathematica doesn't know that it can't take something like

(p[a] p[-a])^-1,

and rewrite it as

p[a]^-1 p[-a]^-1,

which makes no sense. This is probably why your rule is not always working as expected. The expressions it's try to work on sometimes look like

Times[ ..., Power[ p[a], -1], Power[ p[-a], -1], ...],

or along those lines. This does not match the pattern that MakeRule spits out by default.

Using Scalar can help, writing expression instead as

Times ..., Power[ Scalar[ Times[ p[a], p[-a] ] ], -1 ], ...],

and now there is something that matches the pattern returned by MakeRule. ToCanonical tries to add Scalar heads sometimes (I'm not sure when), but I think it doesn't group things in denominators properly for some reason. Try adding Scalar as early as possible when writing your expressions with contractions in the denominator, e.g.

Scalar[ p[a] p[-a] ]^-1,

which will be successfully transformed by the rule you get out of MakeRule.

Leo

Feb 6, 2011, 12:29:15 PM2/6/11

to xAct Tensor Computer Algebra

Hi,

First a general comment. In this expression:

> Expand[Simplification[ContractMetric[dg1, \[Eta]] //. onshell] //.

> onshell //. mand] //. onshell //. mand // Simplification

you use a combination of Expand and Simplification. Note that

Simplification is just Simplify[ ToCanonical[ ... ] ] and therefore

you are spending time in using Simplify and then undoing its action

with Expand. The recommendation is to use ToCanonical at intermediate

steps (which returns expressions expanded), when you think it is

needed, and only at the end use Simplification (or just Simplify if

the expression is already canonicalized).

> But soething like this: Simplification[

> Expand[ContractMetric[dg1, \[Eta]]] //. onshell //. mand]

>

> will leave the contractions in the denominator unreplaced.

As Leo points out, the idea of contraction in the denominator is based

on the assumption that you deal with a scalar. Most functions in

xTensor will not stop to check that, but sometimes ToCanonical

discovers, as part of its standard computation, that this is the case.

Then ToCanonical wraps the expression with Scalar, to help all other

commands knowing this fact. If you work with tensor products in the

denominator, you certainly need to use Scalar. See the functions

PutScalar, BreakScalar, NoScalar. If such denominators are very

frequent, consider the possibility of declaring them, say:

DefTensor[ pp[], M ]

p/: p[a_] p[-a_] := pp[]

xTensor will be faster manipulating pp[] than manipulating

Scalar[ p[a] p[-a] ].

> On a different note, what exactly is SeparateMetric? Is it the

> opposite of ContractMetric?

Essentially yes. The idea of SeparateMetric is the following. We

absorb metric contractions by raising and lowering indices without

changing the stem of the tensor, say g[a,b] T[-b] is written as T[a].

This is simple and useful, but hides the presence of the metric, and

that is potentially dangerous in some computations. What

SeparateMetric does is extracting metric factors from a given tensor

until all its slots have indices are in the positions in which the

tensor was defined. That is, it makes explicit the presence of the

metric field. If you define DefTensor[ T[a,-b], M ] then

SeparateMetric[][ T[a, b] ] will extract a metric factor from the

second slot of T, but not from the first one, etc. ContractMetric

would absorb it again, and in this sense they are inverses. The only

difference is that SeparateMetric takes into account the original

configuration (stored in SlotsOfTensor[ T ]), but ContractMetric

absorbs all metric factors.

> I don't understand its syntax and I get funny indices with numbers when I try it.

SeparateMetric has additional arguments for further flexibility. The

general syntax is SeparateMetric[ metric, basis ][ expr, index ], with

defaults for all arguments except expr. The metric argument specifies

which metrics you want to separate (by default all first-metrics). The

basis argument specifies the type of index used for the new

contraction (by default an abstract index, with formal basis AIndex).

The index argument specifies which indices you want to separate, that

is, the indices already present in the original expression, not the

index added for the new contraction (the default is all indices that

can be separated). The reason why this function has a double pair of

brackets is because I use it frequently with mapping constructs, like

Map[ SeparateMetric[metric], expr , {level}] or

MapAt[ SeparateMetric[metric], expr, position ]. Just that.

Numbered indices, like f1 or f32, are generated when xTensor runs out

of user-defined indices. xTensor takes the last index you declared

with the manifold or vbundle and starts adding numbers sequentially.

If you have a look at IndicesOfVBundle[vbundle] you will see that it

contains two lists: the indices you declared and the indices added by

this mechanism. Note that these are all proper abstract indices,

perfectly usable. They are not dollar-indices (like f$1 or f$32),

which are internal dummies, not to be used directly by users.

ScreenDollarIndices replaces the dollar-dummies with normal indices,

and the latter could be numbered. If you don´t like these, declare

many indices at the beginning.

Cheers,

Jose.

First a general comment. In this expression:

> Expand[Simplification[ContractMetric[dg1, \[Eta]] //. onshell] //.

> onshell //. mand] //. onshell //. mand // Simplification

Simplification is just Simplify[ ToCanonical[ ... ] ] and therefore

you are spending time in using Simplify and then undoing its action

with Expand. The recommendation is to use ToCanonical at intermediate

steps (which returns expressions expanded), when you think it is

needed, and only at the end use Simplification (or just Simplify if

the expression is already canonicalized).

> But soething like this: Simplification[

> Expand[ContractMetric[dg1, \[Eta]]] //. onshell //. mand]

>

> will leave the contractions in the denominator unreplaced.

on the assumption that you deal with a scalar. Most functions in

xTensor will not stop to check that, but sometimes ToCanonical

discovers, as part of its standard computation, that this is the case.

Then ToCanonical wraps the expression with Scalar, to help all other

commands knowing this fact. If you work with tensor products in the

denominator, you certainly need to use Scalar. See the functions

PutScalar, BreakScalar, NoScalar. If such denominators are very

frequent, consider the possibility of declaring them, say:

DefTensor[ pp[], M ]

p/: p[a_] p[-a_] := pp[]

xTensor will be faster manipulating pp[] than manipulating

Scalar[ p[a] p[-a] ].

> On a different note, what exactly is SeparateMetric? Is it the

> opposite of ContractMetric?

absorb metric contractions by raising and lowering indices without

changing the stem of the tensor, say g[a,b] T[-b] is written as T[a].

This is simple and useful, but hides the presence of the metric, and

that is potentially dangerous in some computations. What

SeparateMetric does is extracting metric factors from a given tensor

until all its slots have indices are in the positions in which the

tensor was defined. That is, it makes explicit the presence of the

metric field. If you define DefTensor[ T[a,-b], M ] then

SeparateMetric[][ T[a, b] ] will extract a metric factor from the

second slot of T, but not from the first one, etc. ContractMetric

would absorb it again, and in this sense they are inverses. The only

difference is that SeparateMetric takes into account the original

configuration (stored in SlotsOfTensor[ T ]), but ContractMetric

absorbs all metric factors.

> I don't understand its syntax and I get funny indices with numbers when I try it.

general syntax is SeparateMetric[ metric, basis ][ expr, index ], with

defaults for all arguments except expr. The metric argument specifies

which metrics you want to separate (by default all first-metrics). The

basis argument specifies the type of index used for the new

contraction (by default an abstract index, with formal basis AIndex).

The index argument specifies which indices you want to separate, that

is, the indices already present in the original expression, not the

index added for the new contraction (the default is all indices that

can be separated). The reason why this function has a double pair of

brackets is because I use it frequently with mapping constructs, like

Map[ SeparateMetric[metric], expr , {level}] or

MapAt[ SeparateMetric[metric], expr, position ]. Just that.

Numbered indices, like f1 or f32, are generated when xTensor runs out

of user-defined indices. xTensor takes the last index you declared

with the manifold or vbundle and starts adding numbers sequentially.

If you have a look at IndicesOfVBundle[vbundle] you will see that it

contains two lists: the indices you declared and the indices added by

this mechanism. Note that these are all proper abstract indices,

perfectly usable. They are not dollar-indices (like f$1 or f$32),

which are internal dummies, not to be used directly by users.

ScreenDollarIndices replaces the dollar-dummies with normal indices,

and the latter could be numbered. If you don´t like these, declare

many indices at the beginning.

Cheers,

Jose.

Feb 6, 2011, 4:10:24 PM2/6/11

to xAct Tensor Computer Algebra

Thanks so much to both of you.

For the record I have wrapped the contraction in the denominator when

defining dg1.

Thanks for the comments on efficiency and for the explanation I will

make due note of them.

I have a suggestion to make regarding the documentation: Add what you

just told me about SeparateMetric next to the section on

ContractMetric (it can be very useful for producing Feynman Rules) and

state clearly that Expand is needed to replace contractions.

I can do these additions myself and then send them to you if its most

convenient.

I will keep looking in my notebooks for examples of dodgy

replacements.

Ed

For the record I have wrapped the contraction in the denominator when

defining dg1.

Thanks for the comments on efficiency and for the explanation I will

make due note of them.

I have a suggestion to make regarding the documentation: Add what you

just told me about SeparateMetric next to the section on

ContractMetric (it can be very useful for producing Feynman Rules) and

state clearly that Expand is needed to replace contractions.

I can do these additions myself and then send them to you if its most

convenient.

I will keep looking in my notebooks for examples of dodgy

replacements.

Ed

Feb 28, 2011, 12:04:36 PM2/28/11

to xAct Tensor Computer Algebra

Ok I have another example of something that seems to be a minor bug

but which can be very annoying

defining the usual manifold

onshellk =

MakeRule[{k[a] k[-a], 0}, MetricOn -> All, ContractMetrics -> True];

this

k[a] k[-a] // ToCanonical //. onshellk

just returns k[-a] k[a]

whereas this returns 0 as expected

ToCanonical[k[a] k[-a]] //. onshellk

but which can be very annoying

defining the usual manifold

onshellk =

MakeRule[{k[a] k[-a], 0}, MetricOn -> All, ContractMetrics -> True];

this

k[a] k[-a] // ToCanonical //. onshellk

just returns k[-a] k[a]

whereas this returns 0 as expected

ToCanonical[k[a] k[-a]] //. onshellk

Feb 28, 2011, 1:43:46 PM2/28/11

to Edu Serna, xAct Tensor Computer Algebra

Edu,

This is not an xAct problem. Rather, this has to do with the precedence of different infix operators in Mathematica.

The //. operator (aka ReplaceRepeated) binds more tightly than the // operator (aka Postfix function notation). That means something like

k[a] k[-a] // ToCanonical //. onshellk

is parsed as

k[a] k[-a] // (ToCanonical //. onshellk)

which is not what you intended. Rather, you must tell Mathematica to apply // first, either

(k[a] k[-a] // ToCanonical) //. onshellk

or

ToCanonical[ k[a] k[-a] ] //. onshellk

Hope this helps!

Leo

Feb 28, 2011, 1:52:51 PM2/28/11

to Leo Stein, xAct Tensor Computer Algebra

Thanks so much,

k[a] k[-a] // (ToCanonical //. onshellk) so what does this mean? a replacement being done on ToCAnonical with no argument?

ed

Feb 28, 2011, 1:59:23 PM2/28/11

to Eduardo Serna, xAct Tensor Computer Algebra

That's right. ToCanonical is just a symbol (with some patterns that match for it). You can do ReplaceAll or ReplaceRepeated on anything you want, including "function names" (which are really just symbols). The return value of ( ToCanonical //. onshellk ) will just be ToCanonical, since the replacement has nothing to replace. Then ToCanonical will be applied to the expression to the left.

Leo

Mar 1, 2011, 5:05:03 AM3/1/11

to Leo Stein, xAct Tensor Computer Algebra

Ok thanks so much this is reassuring

Reply all

Reply to author

Forward

0 new messages

Search

Clear search

Close search

Google apps

Main menu