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Carlo Beenakker
Carlo Beenakker
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In quantum physics we sometimes refer to a definition from category theory: Since there is no functorial choice of basis, a proof picks a basis if it defines a non-functorial construction.

In quantum mechanics “picking a basis” is equivalent to assigning sharp values simultaneously to a complete set of observables. For example, this could be the three coordinates $x,y,z$, or it could be the three momentum components $p_x,p_y,p_z$. The obstruction to a functorial choice of basis is expressed by the Kochen-Specker theorem: It is not possible to assign values to all observables. This implies that you cannot avoid picking a basis if you want to describe a measurement.

A problem that appears in the physics context, but does not seem to have a math counterpart, is the "preferred basis paradox". In linear algebra, all bases are equivalent, in our physical reality, they are not. This is dramatized by Schrödinger's cat: the preferred basis is the one were it is either alive or dead, we never observe the cat in a basis obtained by a unitary transformation.

In quantum physics we sometimes refer to definition from category theory: Since there is no functorial choice of basis, a proof picks a basis if it defines a non-functorial construction.

In quantum mechanics “picking a basis” is equivalent to assigning sharp values simultaneously to a complete set of observables. For example, this could be the three coordinates $x,y,z$, or it could be the three momentum components $p_x,p_y,p_z$. The obstruction to a functorial choice of basis is expressed by the Kochen-Specker theorem: It is not possible to assign values to all observables. This implies that you cannot avoid picking a basis if you want to describe a measurement.

A problem that appears in the physics context, but does not seem to have a math counterpart, is the "preferred basis paradox". In linear algebra, all bases are equivalent, in our physical reality, they are not. This is dramatized by Schrödinger's cat: the preferred basis is the one were it is either alive or dead, we never observe the cat in a basis obtained by a unitary transformation.

In quantum physics we sometimes refer to a definition from category theory: Since there is no functorial choice of basis, a proof picks a basis if it defines a non-functorial construction.

In quantum mechanics “picking a basis” is equivalent to assigning sharp values simultaneously to a complete set of observables. For example, this could be the three coordinates $x,y,z$, or it could be the three momentum components $p_x,p_y,p_z$. The obstruction to a functorial choice of basis is expressed by the Kochen-Specker theorem: It is not possible to assign values to all observables. This implies that you cannot avoid picking a basis if you want to describe a measurement.

A problem that appears in the physics context, but does not seem to have a math counterpart, is the "preferred basis paradox". In linear algebra, all bases are equivalent, in our physical reality, they are not. This is dramatized by Schrödinger's cat: the preferred basis is the one were it is either alive or dead, we never observe the cat in a basis obtained by a unitary transformation.
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Carlo Beenakker
Carlo Beenakker
  • 208.3k
  • 19
  • 515
  • 740

In quantum physics we sometimes refer to definition from category theory: Since there is no functorial choice of basis, a proof picks a basis if it defines a non-functorial construction.Since there is no functorial choice of basis, a proof picks a basis if it defines a non-functorial construction.

In quantum mechanics “picking a basis” is equivalent to assigning sharp values simultaneously to a complete set of observables. For example, this could be the three coordinates $x,y,z$, or it could be the three momentum components $p_x,p_y,p_z$. The obstruction to a functorial choice of basis is expressed by the Kochen-Specker theorem: It is not possible to assign values to all observables. This implies that you cannot avoid picking a basis if you want to describe a measurement.

A problem that appears in the physics context, but does not seem to have a math counterpart, is the "preferred basis paradox". In linear algebra, all bases are equivalent, in our physical reality, they are not. This is dramatized by Schrödinger's cat: the preferred basis is the one were it is either alive or dead, we never observe the cat in a basis obtained by a unitary transformation.

In quantum physics we sometimes refer to definition from category theory: Since there is no functorial choice of basis, a proof picks a basis if it defines a non-functorial construction.

In quantum mechanics “picking a basis” is equivalent to assigning sharp values simultaneously to a complete set of observables. For example, this could be the three coordinates $x,y,z$, or it could be the three momentum components $p_x,p_y,p_z$. The obstruction to a functorial choice of basis is expressed by the Kochen-Specker theorem: It is not possible to assign values to all observables.

In quantum physics we sometimes refer to definition from category theory: Since there is no functorial choice of basis, a proof picks a basis if it defines a non-functorial construction.

In quantum mechanics “picking a basis” is equivalent to assigning sharp values simultaneously to a complete set of observables. For example, this could be the three coordinates $x,y,z$, or it could be the three momentum components $p_x,p_y,p_z$. The obstruction to a functorial choice of basis is expressed by the Kochen-Specker theorem: It is not possible to assign values to all observables. This implies that you cannot avoid picking a basis if you want to describe a measurement.

A problem that appears in the physics context, but does not seem to have a math counterpart, is the "preferred basis paradox". In linear algebra, all bases are equivalent, in our physical reality, they are not. This is dramatized by Schrödinger's cat: the preferred basis is the one were it is either alive or dead, we never observe the cat in a basis obtained by a unitary transformation.
Source Link
Carlo Beenakker
Carlo Beenakker
  • 208.3k
  • 19
  • 515
  • 740

In quantum physics we sometimes refer to definition from category theory: Since there is no functorial choice of basis, a proof picks a basis if it defines a non-functorial construction.

In quantum mechanics “picking a basis” is equivalent to assigning sharp values simultaneously to a complete set of observables. For example, this could be the three coordinates $x,y,z$, or it could be the three momentum components $p_x,p_y,p_z$. The obstruction to a functorial choice of basis is expressed by the Kochen-Specker theorem: It is not possible to assign values to all observables.