Translation invariance and the Wightman axioms

[crowlogic]

Translation invariance in the Wightman axioms of quantum field theory is embedded in the formal structure and properties of the fields and states. Here’s how it specifically appears:

Wightman Axioms and Translation Invariance

1. Axiom of Translation Invariance (Poincaré Invariance):

   • Formal Statement: There exists a unitary representation ( U(a) ) of the translation group on the Hilbert space of states such that for any spacetime point ( a ) and any field ( \phi(x) ), [ U(a) \phi(x) U(a)^{-1} = \phi(x + a). ]
   • Meaning: This ensures that the field operators ( \phi(x) ) transform correctly under translations in spacetime. It implies that the physical laws described by these fields are the same regardless of where or when they are applied, reflecting the homogeneity of spacetime.

2. Spectrum Condition:

   • Formal Statement: The joint spectrum of the energy-momentum operators ( P^\mu ) (the generators of translations) lies within the forward light cone ( V^+ ), i.e., the eigenvalues ( p^\mu ) of ( P^\mu ) satisfy ( p^0 \geq 0 ) and ( p^\mu p_\mu \geq 0 ).
   • Meaning: This condition enforces that physical states have non-negative energy and respects the causality structure of spacetime. It ensures that translations in time (evolution) and space (momentum) align with relativistic constraints.

3. Cyclic Vacuum and Translation Invariance:

   • Formal Statement: The vacuum state ( |0\rangle ) is invariant under translations: [ U(a) |0\rangle = |0\rangle. ]
   • Meaning: The vacuum state does not change when translated in spacetime, reflecting the idea that it is the same everywhere and at all times, a foundational requirement for a stable ground state in QFT.

4. Wightman Functions and Translation Invariance:

   • Formal Statement: The Wightman functions (n-point correlation functions) ( W_n(x_1, \ldots, x_n) = \langle 0 | \phi(x_1) \ldots \phi(x_n) | 0 \rangle ) satisfy translation invariance: [ W_n(x_1 + a, \ldots, x_n + a) = W_n(x_1, \ldots, x_n). ]
   • Meaning: The correlation functions, which encode the fundamental interactions and observables in the theory, are invariant under spacetime translations, ensuring consistency with the principle that physical observations do not depend on absolute spacetime positions.

Summary

Translation invariance in the Wightman axioms ensures that the theory respects the fundamental symmetries of spacetime, leading to conservation laws and the proper behavior of physical states and observables. These properties are crucial for the consistency and physical relevance of quantum field theories.

[crowlogic]

Translation invariance in the context of the Wightman axioms of quantum field theory manifests in several key ways:

1. Poincaré Invariance: The Wightman axioms include invariance under the Poincaré group, which encompasses translations in space and time. Specifically, this is expressed through the existence of a strongly continuous unitary representation ( U(a, \Lambda) ) of the Poincaré group acting on the Hilbert space of states. The translation part of the Poincaré group ensures that for any space-time translation ( a ), the field operators transform as: [ U(a) \phi(x) U(a)^{-1} = \phi(x + a). ] This axiom reflects the fundamental symmetry that the laws of physics are the same at all points in space-time.

2. Spectral Condition: The spectral condition, or spectrum condition, is an axiom that requires the joint spectrum of the energy-momentum operators ( P^\mu ) (which generate translations in space-time) to lie within the forward light cone. This means that the four-momentum ( p^\mu ) associated with any physical state satisfies: [ p^0 \geq 0, \quad p^2 = (p^0)^2 - \vec{p}^2 \geq 0. ] This condition ensures that translations do not lead to non-physical states with negative energy.

3. Locality: While not directly translation invariance, locality in the Wightman axioms states that field operators at spacelike separated points commute or anticommute, depending on whether the fields are bosonic or fermionic. The translation invariance part of the Poincaré group ensures that this commutation relation is preserved under translations, maintaining the causal structure of the theory.

Translation invariance in the Wightman axioms is thus integral to maintaining the consistency and physicality of quantum field theories by ensuring that the theory respects fundamental symmetries of space-time and that physical laws are uniformly applicable everywhere.

[crowlogic]

subsume within https://github.com/crowlogic/arb4j/issues/468