berezin_measure
========================================

.. py:class:: grassmanntn.arith.berezin_measure(var_list, reverse=False)

.. py:function:: grassmanntn.arith.d(var_list, reverse=False)

An object representing a Berezin integral operator.

**Parameters:**

- **var_list:** `grassmann_number <grassmann_number.html>`__ or a list of `grassmann_number <grassmann_number.html>`__

  The variable(s) to be integrated.

- **reverse:** `True` or `False` (default)

  If False, the measure of each variables is ordered from left to right. If True, from right to left. If `var_list` is not a list, this option does not matter.

**Returns:**

- **out:** `berezin_measure <berezin_measure.html>`__

  The associated integration measure.


Attributes
++++++++++

Binary operations
+++++++++++++++++++++++++++

- **Multiplication**: `<self>*<other>` and `<other>*<self>`
  
  If multiply to a `grassmann_number <grassmann_number.html>`__, returns an integration.
  If multiply to another `berezin_measure <berezin_measure.html>`__, returns a combined integration measure.



Examples
++++++++

Consider the following Grassmann number.

.. code-block:: python

  >>> import numpy as np
  >>> from grassmanntn import arith as arith
  >>> 
  >>> ψ = arith.set_ac(["ψ1","ψ2"])
  >>> φ = arith.set_ac(["φ1","φ2"])
  >>> K = np.array( [[3,7],[2,5]] )
  >>> W = arith.exp( - φ @ K @ ψ )
  >>> W
  1.0 + (-3.0)*φ1^ψ1 + (-2.0)*φ2^ψ1 + (-7.0)*φ1^ψ2 + (-5.0)*φ2^ψ2 + (-1.0)*φ1^φ2^ψ1^ψ2

We define the ψ-integral and the φ-integral.

.. code-block:: python

  >>> # multivariable measure
  >>> dψ = arith.d(ψ)
  >>> dψ
  ∫ψ1 ∫ψ2
  >>> dφ = arith.d(φ)
  >>> dφ
  ∫φ1 ∫φ2 

We can apply the integral by multiplying the measure directly to the Grassmann number:

.. code-block:: python

  >>> dψ*W
  (1.0)*φ1^φ2
  >>> dφ*W
  (1.0)*ψ1^ψ2

The product is associative.

.. code-block:: python
  
  >>> dψ*dφ*W
  -1.0
  >>> dψ*(dφ*W)
  -1.0
  >>> (dψ*dφ)*W
  -1.0