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Symmetries of free massless particles and soft theorems

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Abstract

In an earlier paper we have constructed a basis of massless single particle quantum states which transform in the unitary principal series representation of the four dimensional Lorentz group. The S-matrix written in this basis gives rise to the Mellin transformed amplitude of Pasterski–Shao–Strominger and its generalization. In this basis the particle can be thought of as living on the null-infinity in the Minkowski space. In this paper we take some preliminary steps to see how the connection between soft theorems and symmetries work out in this picture. For simplicity we consider only the leading soft photon and soft graviton theorems which are related to U(1) Kac–Moody and supertranslations.

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Notes

  1. The construction of free quantum fields on the abstract \((u,z,\bar{z})\) space is naturally done from a group theoretic point of view [23] by following Wigner. This construction does not require the knowledge of the (Minkowski) space–time quantum fields. For massless particles one can construct the delta function normalisable states,

    figure a

    where \(|\omega ,z,\bar{z},\sigma>\) are the standard Wigner states. Now one can define the Heisenberg picture states,

    figure b

    where \(u = U(1+z\bar{z})\) and \(H=P^0\) is the Hamiltonian. These states are analogous to the states \(|{\vec {x}}, t> = e^{iHt}|{\vec {x}}>\) in the non-relativistic quantum mechanics. Now, one can show [23] that the states \(|\lambda ,\sigma ,u,z,\bar{z}>\) transform covariantly under the full Poincare group. The transformation laws are given in Eqs.—(3.9), (3.10), (3.12) and (3.13). The transformation laws suggest that we can think of the particle described by the state \(|\lambda ,\sigma ,u,z,\bar{z}>\) as located at the point \((u,z,\bar{z})\). The definitions (3.4) and (3.5) are essentially the rewriting of the Eq. (3.5) using creation–annihilation operators.

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Acknowledgements

I would like to thank especially Ashoke Sen for numerous enlightening discussions on soft theorems and related subjects. I would also like to thank Alok Laddha and Prahar Mitra for very helpful discussion on soft theorems and related matters. Part of this work was presented in the “First Spring Meeting on Strings” held in NISER Bhubaneswar, India. I would like to thank the organizers for holding this exciting meeting. I would also like to thank all the participants of the meeting especially Jyotirmoy Bhattacharya, Sayantani Bhattacharya, Bobby Ezhuthachan, Arnab Kundu, Jnan Maharana, Sudhakar Panda, Ashoke Sen and Yogesh Srivastava for their valuable feedback.

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  1. Institute of Physics, Sachivalaya Marg, Bhubaneshwar, 751005, India

    Shamik Banerjee

  2. Homi Bhabha National Institute, Anushakti Nagar, Mumbai, 400085, India

    Shamik Banerjee

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Banerjee, S. Symmetries of free massless particles and soft theorems. Gen Relativ Gravit 51, 128 (2019). https://doi.org/10.1007/s10714-019-2609-z

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