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## /minkowski [ Modules ]

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NAME

module minkowski

PURPOSE

This module defines functions which are connected to Relativity: Metric Tensor, Scalar Product, Gamma matrices, ...

NOTES

• Uses the "mostly -" metric.

## minkowski/gamma [ Global module-variables ]

NAME

complex, dimension(0:3,0:3,0:11), parameter :: gamma

PURPOSE

Represents the gamma matrices gamma0-gamma3, gamma5, gamma6-gamma11.

## minkowski/metricTensor [ Global module-variables ]

PURPOSE

The metric tensor

SOURCE

  real, dimension(0:3,0:3), public, parameter :: metricTensor = reshape((/ 1., 0., 0., 0., &
0.,-1., 0., 0., &
0., 0.,-1., 0., &
0., 0., 0.,-1. /),(/4,4/))


## minkowski/contract [ Subroutines ]

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NAME

real function Contract(a,b)

PURPOSE

Evaluates a^(mu nu) b_(mu nu)

INPUTS

• a,b : matrices a^(mu nu) b^(mu nu)
• The matrices can be real or complex (therefore we use an interface)

OUTPUT

• real

## minkowski/abs4 [ Functions ]

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NAME

real function abs4(a) real function abs4(a, flagOk)

PURPOSE

Absolute value of a 4-vector.

INPUTS

• real,dimension(0:3) :: a

OUTPUT

• real :: abs4=sqrt(a(0)*a(0)-a(1)*a(1)-a(2)*a(2)-a(3)*a(3))

## minkowski/op_ang [ Functions ]

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NAME

real function op_ang(p1,p2)

PURPOSE

Computes the opening angle between the spatial components of two 4-vectors.

INPUTS

• real,dimension(0:3) :: p1,p2

OUTPUT

• opening angle in degrees [0...180]

## minkowski/abs3 [ Functions ]

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NAME

function abs3(a)

PURPOSE

Absolute value of the spatial components of a 4-vector.

INPUTS

• real,dimension(0:3) :: a

OUTPUT

• real :: abs3=sqrt(a(1)**2+a(2)**2+a(3)**2)

## minkowski/abs4Sq [ Functions ]

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NAME

real function abs4Sq(a)

PURPOSE

Absolute value squared of a 4-Vector.

INPUTS

• real,dimension(0:3) :: a ! four vector

OUTPUT

• real :: abs4Sq=a(0)*a(0)-a(1)*a(1)-a(2)*a(2)-a(3)*a(3)

## minkowski/SP [ Functions ]

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NAME

function SP(a,b)

PURPOSE

Scalar Product for 4-Vectors, "mostly -" metric

INPUTS

• real,dimension(0:3) :: a,b ! four vectors

OUTPUT

• real :: SP=a(0)*b(0)-a(1)*b(1)-a(2)*b(2)-a(3)*b(3)

## minkowski/ContractCC [ Functions ]

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NAME

real function ContractCC(a,b)

PURPOSE

Evaluates a^(mu nu) b_(mu nu)

INPUTS

• complex,dimension(0:3,0:3) :: a,b ! matrices a^(mu nu) b^(mu nu)

OUTPUT

• real

## minkowski/ContractRR [ Functions ]

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NAME

real function ContractRR(a,b)

PURPOSE

Evaluates a^(mu nu) b_(mu nu)

INPUTS

• real,dimension(0:3,0:3) :: a,b ! matrices a^(mu nu) b^(mu nu)

OUTPUT

• real

## minkowski/sigma4 [ Functions ]

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NAME

complex function sigma4(a)

PURPOSE

Returns sigma^(mu nu)=i/2 [gamma^mu, gamma^nu]

INPUTS

OUTPUT

• complex, dimension(0:3,0:3) :: matrix

## minkowski/slashed [ Functions ]

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NAME

function slashed(p) result(matrix)

PURPOSE

Evaluates gamma^mu*p_mu

INPUTS

• real, dimension(0:3) :: p

OUTPUT

• complex, dimension(0:3,0:3) :: matrix

## minkowski/slashed5 [ Functions ]

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NAME

function slashed5(p) result(matrix)

PURPOSE

Evaluates gamma^mu*p_mu*gamma_5

INPUTS

• real, dimension(0:3) :: p

OUTPUT

• complex, dimension(0:3,0:3) :: matrix

## minkowski/tilde [ Functions ]

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NAME

function tilde(a) result(a_tilde)

PURPOSE

Evaluates a^tilde=gamma_0 a^dagger gamma_0

INPUTS

• complex, dimension(0:3,0:3) :: a

OUTPUT

• complex, dimension(0:3,0:3) :: a_tilde

## minkowski/levi_civita [ Functions ]

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NAME

integer function levi_civita (i, j, k, l)

PURPOSE

Calculates the fully antisymmetric Levi-Civita tensor \epsilon_{ijkl} in four dimensions.

INPUTS

• integer :: i, j, k, l --- tensor indices

OUTPUT

• tensor value for the given indices