Physics[FeynmanIntegral][TensorBasis] - compute a basis of tensor structures from a given list of external momentum and another one with free spacetime indices
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Calling Sequence
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TensorBasis(list_of_external_momenta, list_of_spacetime_indices)
TensorBasis(list_of_external_momenta, list_of_spacetime_indices, symmetrize = ..)
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Parameters
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list_of_external_momenta
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a list of external momenta, which by convention in the FeynmanIntegral package are written as P__n where n is an integer
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list_of_spacetime_indices
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a list of spacetime indices, that could be covariant or contravariant (preceded by )
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symmetrize = ..
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(optional) the right-hand side can be true (default) or false, to symmetrize the products of external momenta that appear in the returned basis
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Description
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TensorBasis receives a list of external momenta, which by convention in the FeynmanIntegral package are written as P__n where n is an integer, and a list of spacetime indices, which by default are represented by greek letters (to change the kind of letter see Setup) and returns a tensor basis onto which one can expand a tensorial structure with as many indices as in list_of_spacetime_indices.
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The tensor basis returned is constructed by taking the multiple-Cartesian product of the list of external momenta, and the metric , as many times as the number of indices in the list of spacetime indices, and discarding permutations.
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The tensor basis is returned symmetrized, e.g. if a product of two tensors appears in the basis, then the output contains . To receive the tensor basis non-symmetrized pass the optional argument symmetrize = false
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These tensor basis are relevant in the context of the Passarino-Veltman approach for the reduction of tensor to scalar Feynman integrals implemented in the TensorReduce command.
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Examples
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To remain closer to textbook notation, display the imaginary unit with a lowercase
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interface(imaginaryunit = i):
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The simplest case is that of a single external momentum and only one spacetime index
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TensorBasis([P__1], [mu]);
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This basis allows for expressing the following tensor Feynman integral as a linear combination of the elements of the basis
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%FeynmanIntegral(p__1[~mu]/((p__1^2 - m__phi^2 + i * epsilon)*((p__1 - P__1)^2 - m__1^2 + i * epsilon)), p__1);
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TensorReduce((3), step = 1);
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opening the way for the reduction process
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(3) = TensorReduce((3));
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and ultimately leading to its symbolic computation by evaluating the scalar FeynmanIntegrals above
The case of two spacetime indices already results in a basis even when there are no external momenta
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TensorBasis([], [mu, nu]);
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Products of the metric are introduced when the number of indices makes that necessary
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TensorBasis([], [mu, nu, alpha, beta]);
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The non-symmetrized form of this basis
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TensorBasis([], [mu, nu, alpha, beta], symmetrize = false);
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Two more realistic examples
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TensorBasis([P__1, P__2, P__3], [mu, nu]);
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TensorBasis([P__1], [mu, nu, alpha]);
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| (11) |
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See Also
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Dgamma, Evaluate, FeynmanDiagrams, FeynmanIntegral[Overview], FromAbstractRepresentation, Parametrize, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup, TensorReduce, ToAbstractRepresentation
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References
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[1] Smirnov, V.A., Feynman Integral Calculus. Springer, 2006.
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[2] Weinberg, S., The Quantum Theory Of Fields. Cambridge University Press, 2005.
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[3] Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982.
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Compatibility
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The Physics[FeynmanIntegral][TensorBasis] command was introduced in Maple 2021.
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