Interchaging boson and fermion on an infinite 1 dimensional line

Question

In 1+1 dim bosonization, one introduce the Klein factors, which are Hermitian and satisfies Clifford algebra.

(1) In the case of 1 dim space is a 1D ring ($S^1$ circle), then one have left-right boson field commutes $$[\phi_L, \phi_R]=0$$ but introduces Klein factor to reproduce the fermionized fermion field anti-commute: $$\{ \psi_L, \psi_R\}=0.$$

(2) However, according to this Ref, in page 21, footnote 8, for 1 dim space as an infinite line, one requires $$[\phi_L, \phi_R]=i \frac{1}{4}$$ to reproduces the fermionized fermion field anti-commute: $$\{ \psi_L, \psi_R\}=0.$$

How can I see, how can one show that $[\phi_L, \phi_R]=i \frac{1}{4}$ for 1 dim space as an infinite line?

Answer 1

If you work with the dual fields $\varphi$ and $\vartheta$ (where $\partial_x\vartheta=-\frac{1}{v}\partial_t\varphi$), you see that their commutation relation gives $-i\theta(x-x^\prime)$ at equal time as shown in eq.(39) of the same reference you have provided. Since left- and right-modes are just the linear combinations $\phi_L=\frac{1}{2}(\varphi+\vartheta)$ and $\phi_R=\frac{1}{2}(\varphi-\vartheta)$, you see that indeed $$ [\phi_L(t,x_1),\phi_R(t,x_2)]=\frac{i}{4}\,. $$

This is for the whole 1D line. The difference with a periodic space of with a space with a boundary is that the integration of the duality condition $\partial_x\vartheta=-\frac{1}{v}\partial_t\varphi$ is sensitive to the boundary conditions for the dual field. In any case, this is not a big deal since the Klein factors $\eta$ and $\tilde{\eta}$ can be built in terms of the bosonic fields at the boundary anyway. They are something of the sort $\eta\sim \mathrm{exp}[i\alpha Q]$, $\tilde{\eta}\sim \mathrm{exp}[i\beta\tilde{Q}]$ where $Q$ and $\tilde{Q}$ are the ``charges'' $$ Q\sim \int dx \partial_t\varphi \,,\qquad \tilde{Q}\sim \int dx \partial_t\vartheta $$ associated to the symmetries $\varphi\rightarrow\varphi+c$, $\vartheta\rightarrow\vartheta+\tilde{c}$ that are broken by the boundary conditions. By duality, these Klein factors are proportional to the values of $\vartheta_{\mathrm{boundary}}$ and $\varphi_{\mathrm{boundary}}$ which can be used to adjust the commutation relations as you please (by multiplying the exponential of $\varphi$ and $\vartheta$ that you use to bosonize by those Klein factors).