koszul tate resolution of cohomology for a free qed/maxwell theory
22 Jan, 12:18

The Koszul-Tate resolution is a powerful tool in the BRST (Becchi-Rouet-Stora-Tyutin) formalism used to handle gauge theories, including free quantum electrodynamics (QED) or Maxwell theory. This resolution helps in understanding the cohomology of the BRST differential, which is crucial for identifying physical observables and anomalies in the theory.

For a free QED/Maxwell theory, the Koszul-Tate resolution can be constructed as follows:

  1. Field Content:

    • Gauge field: AμA_\mu
    • Ghost field: cc (associated with gauge symmetry)
    • Antifields: A∗μA^{*\mu}, c∗c^*

  2. BRST Transformations:

    • sAμ=∂μcsA_\mu = \partial_\mu c
    • sc=0sc = 0
    • sA∗μ=∂νFνμsA^{*\mu} = \partial_\nu F^{\nu\mu} (where Fνμ=∂νAμ−∂μAνF^{\nu\mu} = \partial^\nu A^\mu - \partial^\mu A^\nu is the field strength)
    • sc∗=−∂μA∗μsc^* = -\partial_\mu A^{*\mu}

  3. Koszul-Tate Differential: The Koszul-Tate differential δ\delta acts on the antifields and is defined by:

    • δA∗μ=∂νFνμ\delta A^{*\mu} = \partial_\nu F^{\nu\mu}
    • δc∗=−∂μA∗μ\delta c^* = -\partial_\mu A^{*\mu}

  4. Cohomology: The cohomology of the Koszul-Tate differential δ\delta in the space of local functionals is trivial in positive antifield number. This means that any δ\delta-closed functional with positive antifield number is also δ\delta-exact.

  5. Resolution: The Koszul-Tate resolution provides a way to systematically construct the BRST cohomology. For free QED, the resolution involves the following complex:

0⟶Functions of (Aμ,c)→δFunctions of (Aμ,c,A∗μ)→δFunctions of (Aμ,c,A∗μ,c∗)⟶0 0 \longrightarrow \text{Functions of } (A_\mu, c) \xrightarrow{\delta} \text{Functions of } (A_\mu, c, A^{*\mu}) \xrightarrow{\delta} \text{Functions of } (A_\mu, c, A^{*\mu}, c^*) \longrightarrow 0

Here, the arrows represent the action of the Koszul-Tate differential δ\delta.

  1. Physical Observables: The physical observables are represented by the cohomology classes of the BRST differential ss at ghost number zero. These are the gauge-invariant functionals of the fields that are not ss-exact.

  2. Anomalies: Anomalies correspond to the cohomology classes of the BRST differential ss at ghost number one. For free QED, there are no anomalies, meaning the cohomology at ghost number one is trivial.

In summary, the Koszul-Tate resolution for a free QED/Maxwell theory involves constructing a complex where the Koszul-Tate differential δ\delta acts on the antifields. The cohomology of this differential helps in identifying physical observables and ensuring the absence of anomalies in the theory.

22 Jan, 12:18

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