1. Natsumi Nagata E-ken Journal Club December 21, 2012 Minimal fields of canonical dimensionality are free S. Weinberg, Phys. Rev. D86, 105015 (2012) [1210.3864].

2. 1. Introduction

3. Statement In a scale-invariant relativistic field theory, any fields that belong to the (j, 0) or (0, j) representations of the Lorentz group have canonical dimensionality d = j+1 are necessarily free !! This conclusion is already known for j=0. D. Buchholz and K. Fredenhagen, J. Math. Phys. 18, 1107 (1977). Here, it is extended to all spins.

4. Theorem Although conformal invariance is not used here, this result gains interest once you consider the following theorem: In a conformal theory, the only fields that can describe massless particles belong to the (j, 0) or (0, j) representations of the Lorentz group have canonical dimensionality d = j+1 G. Mack, Commun. Math. Phys. 55, 1 (1977).

5. Conclusion These two results indicate that In conformal field theories massless particles must be free particles. In conformal field theories massless particles must be free particles.

6. Preliminary Lorentz transformation Infinitesimal Lorentz transformation Rotation & boost A, B-matrices

7. Preliminary Commutation relations SU(2) algebra The representation of the Lorentz group is labeled by the values of the positive integers and/or half-integers A & B. We discuss the case where (A, B) = (j, 0) or (0, j). Dilatation We consider the case where d = j + 1. d: scaling dimension

8. 2. Proof (main result)

9. Let us consider a field φ n (x) belonging to any representation of the Lorentz group and the vacuum expectation value Translational invariance Lorentz transformation It follows that and its infinitesimal transformation leads to

10. with L ρσ the differential operator and [ J ρσ ] nm are the matrix representation of the Lorentz generators. Iteration of the above equation leads to On the other hand, where S is the scale transformation operator

11. If the field φ n (x) belongs to the (j, 0) representation of the Lorentz group, Then, Also,

12. Further, Substituting the results into the equation, we obtain

13. Scale transformation The scale transformation property of φ n (x) is Then, in a scale-invariant theory, So, we have As a result,

14. In particular, for d = j + 1, So, we obtain

15. Lemma Any local operator that annihilates the vacuum must vanish. R. Jost (19??), B. Schroer (19??). P. G. Federbush and K. A. Johnson, Phys. Rev. 120, 1926 (1960). A proof based on the axiomatic field theory approach is given in F. Strocchi, Phys. Rev. D17, 2012 (1978). By using the result, we eventually derive and the field is therefore free. The proof for (0, j) fields is identical.

16. Remarks The above statement does not say that the theory is a free-field theory. There may be other fields in the same theory, which transform according to other representations of the Lorentz group and/or have other dimensionalities.

17. Remarks The proof does not assume the existence of a Lagrangian. A theory can be given by a set of correlators. If a theory has a Lagrangian, it is crucial for the proof to assume Lorentz and scale invariance of both the Lagrangian and the vacuum. In this case, if the scale invariance is spontaneously broken, (j, 0)- or (0, j)-fields with d=j+1 are not necessarily free. A. Monin, and M. Shaposhnikov, arXiv: 1211.3543 [hep-th].

18. 3. Proof (Federbush-Johnson theorem) F. Strocchi, Phys. Rev. D17, 2012 (1978).

19. Axiom The physical Hilbert space H is spanned by Φ i (x i ) : a real field f i (x i ) : an arbitrary function (test functions) |0> : the vacuum (Generalization for generic fields is trivial.) In what follows, we use the following notation: Cyclicity of the vacuum smeared fields

20. Reeh-Schlieder’s theorem O : an open set of space-time Let F ( O ) the set of polynomials in the fields φ i [f i ], smeared with test functions f i having support contained in O. Then, { F ( O )|0>} also spans the Hilbert space H. O

21. Proof. Let us assume that there exists a vector |ψ> in H such that Then, for any functions having support in O, So, and the analytic property of leads to (the edge of the wedge theorem)

22. Federbush-Johnson’s theorem O : an open set of space-time O ’ = { x μ | (x – y) 2 < 0 for all y in O } If O ’ is not empty and then, Proof. If for then, for any T Since F ( O ’ )|0> spans H, we conclude O O’O’

23. Intuitive demonstration (Reeh-Schlieder theorem)

24. 4. Proof (Mack’s theorem)

25. Theorem In a conformal theory, the only fields that can describe massless particles belong to the (j, 0) or (0, j) representations of the Lorentz group have canonical dimensionality d = j+1 G. Mack, Commun. Math. Phys. 55, 1 (1977).

26. One-particle states for massless particles To define the massless particle states, we first introduce a standard three-momentum: Little group Infinitesimal transformation with

27. By using the commutation relations the little group is ISO(2) (Rotations and translations in the 2-dimensions.) we have I2I2 I1I1 J3J3

28. This assumption leads to The states are then distinguished by the eigenvalues of the remaining generator The little group has an invariant Abelian subgroup In order to avoid introducing new continuous degrees of freedom, it is necessary to assume that the generators of the invariant Abelian subgroup annihilate the states. We then take with L μ ν (p) that takes k to p.

29. As we have seen above, the representation of the little group is in general parameterized as Then, So, the Lorentz transformation does not change the helicity Extra phase factor φ(p, Λ)

30. Performing the infinitesimal Lorentz transformation, we have where, Further, by taking a particular choice for L μ ν (p), we fix

31. Conformal algebra S: scale transformation, K μ : special conformal transformation Poincare algebra

32. These commutation relations are satisfied by the following operators on one-particle states: In fact, these transformation rules are unique.

33. Transformation rules of fields (Note that this excludes gauge fields) We will consider here only fields that transform linearly and homogeneously under Poincare transformations d n : scaling dimension

34. By a field ``describing” a particle, we mean that the field has non-vanishing matrix elements between the particle state and the vacuum. The scale transformation property of the fields then gives scale as p A+B d = A + B + 1 S. Weinberg, Phys. Rev. 134, B882 (1964).

35. Now let us consider special conformal transformations. After a lengthy computation, we obtain

36. But by using the relations, we see that therefore

37. The only irreducible representations for which J 2 takes the same value for all components are the (j, 0) and (0, j) representations In this case, the scaling dimensions of these fields are d = j+ 1

38. Conclusion Now we conclude that In conformal field theories massless particles must be free particles. In conformal field theories massless particles must be free particles.

39. Backup

40. Prescription for obtaining the expression of the generators in conformal theories operating on the one-particle states Prescription for obtaining the expression of the generators in conformal theories operating on the one-particle states ① where α(p) is a real function, and we have used

41. ②

42. ③ is a function of |p|.

43. ④ ⑤