LSZ reduction formula

In quantum field theory, the LSZ reduction formula is a method to calculate S-matrix elements (the scattering amplitudes) from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.

Although the LSZ reduction formula cannot handle bound states, massless particles and topological solitons, it can be generalized to cover bound states, by use of composite fields which are often nonlocal. Furthermore, the method, or variants thereof, have turned out to be also fruitful in other fields of Theoretical Physics. For example in Statistical Physics they can be used to get a particularly general formulation of the Fluctuation-dissipation theorem.

In and Out fields

S-matrix elements are amplitudes of transitions between "in" states and "out" states. An "in" state $$|{p} mathrm{in} angle describes the state of a system of particles which, in a far away past, before interacting, were moving freely with definite momenta $${p}, and, conversely, an "out" state $$|{p} mathrm{out} angle describes the state of a system of particles which, long after interaction, will be moving freely with definite momenta $${p}.

"In" and "out" states are states in Heisenberg picture so they should not be thought to describe particles at a definite time, but rather to describe the system of particles in its entire evolution, so that the S-matrix element:

:$$S_{fi}=langle {q} mathrm{out}| {p} mathrm{in} angle

is the probability amplitude for a set of particles which were prepared with definite momenta $${p} to interact and be measured later as a new set of particles with momenta $${q}.

The easy way to build "in" and "out" states is to seek appropriate field operators that provide the right creation and annihilation operators. These fields are called respectively "in" and "out" fields.

Just to fix ideas, suppose we deal with a Klein-Gordon field that interacts in some way which doesn't bother us:

:$$mathcal L= frac 1 2 part_mu varphipart^mu varphi - frac 1 2 m_0^2 varphi^2 +mathcal L_{mathrm{int

$$mathcal L_{mathrm{int may contain a self interaction $$g varphi^3 or interaction with other fields, like a Yukawa interaction $$g varphiarpsipsi. From this lagrangian, using Euler-Lagrange equations, the equation of motion follows:

:$$left(part^2+m_0^2 ight)varphi(x)=j_0(x)

where, if $$mathcal L_{mathrm{int does not contain derivative couplings:

:$$j_0=frac{partmathcal L_{mathrm{int}{part varphi}

We may expect the "in" field to resemble the asymptotic behaviour of the interacting field as $$x^0 ightarrow-infty, making the assumption that in the far away past interaction described by the current $$j_0 is neglegible, as particle are far from each other. This hypothesis is named 'adiabatic hypothesis'. However self interaction never fades away and, besides many other effects, it causes a difference between the lagrangian mass $$m_0 and the physical mass $$m of the $$varphi boson. This fact must be taken into account by rewriting the equation of motion as follows:

:$$left(part^2+m^2 ight)varphi(x)=j_0(x)+left(m^2-m_0^2 ight)varphi(x)=j(x)

This equation can be solved formally using the retarded Green's function of the Klein-Gordon operator $$part^2+m^2:

:$$Delta_{mathrm{ret(x)=i hetaleft(x^0 ight)int frac{mathrm{d}^3k}{(2pi)^3 2omega_k} left(e^{-ikcdot x}-e^{ikcdot x} ight)_{k^0=omega_k};quadomega_k=sqrt{mathbf{k}^2+m^2}

allowing us to split interaction from asymptotic behaviour. The solution is:

:$$varphi(x)=sqrt Z varphi_{mathrm{in(x) +int mathrm{d}^4y Delta_{mathrm{ret(x-y)j(y)

The factor $$sqrt Z is a normalization factor that will come handy later, the field $$varphi_{mathrm{in is a solution of the homogeneous equation associated with the equation of motion:

:$$left(part^2+m^2 ight) varphi_{mathrm{in(x)=0,

and hence is a free field which describes an incoming unperturbed wave, while the last term of the solution gives the perturbation of the wave due to interaction.

The field $$varphi_{mathrm{in is indeed the "in" field we were seeking, as it describes the asymptotic behaviour of the interacting field as $$x^0 ightarrow-infty, though this statement will be made more precise later . It is a free scalar field so it can be expanded in flat waves:

:$$varphi_{mathrm{in(x)=int mathrm{d}^3k left{f_k(x) a_{mathrm{in(mathbf{k})+f^*_k(x) a^dagger_{mathrm{in(mathbf{k}) ight}

where::$$f_k(x)=left.frac{e^{-ikcdot x{(2pi)^{3/2}(2omega_k)^{1/2 ight|_{k^0=omega_k}

The inverse function for the coefficients in terms of the field can be easily obtained and put in the elegant form::$$a_{mathrm{in(mathbf{k})=iint mathrm{d}^3x f^*_k(x)overleftrightarrowpart_0varphi_{mathrm{in(x);quadoverleftrightarrowpart_0=overrightarrowpart_0-overleftarrowpart_0

The Fourier coefficients satisfy the algebra of creation and annihilation operators:

:$$ [a_{mathrm{in(mathbf{p}),a_{mathrm{in(mathbf{q})] =0;quad [a_{mathrm{in(mathbf{p}),a^dagger_{mathrm{in(mathbf{q})] =delta^3(mathbf{p}-mathbf{q});

and they can be used to build "in" states in the usual way:

:$$left|k_1,ldots,k_n mathrm{in} ight angle=sqrt{2omega_{k_1a_{mathrm{in^dagger(mathbf{k}_1)ldots sqrt{2omega_{k_na_{mathrm{in^dagger(mathbf{k}_n)|0 angle

The relation between the interacting field and the "in" field is not very simple to use, and the presence of the retarded Green's function tempts us to write something like:

:$$varphi(x)simsqrt Zvarphi_{mathrm{in(x)quad mathrm{as}quad x^0 ightarrow-infty

implicitly making the assumption that all interactions become neglegible when particles are far away from each other. Yet the current $$j(x) contains also self interactions like those producing the mass shift from $$m_0 to $$m. These interactions do not fade away as particles drift apart, so much care must be used in establishing asymptotic relations between the interacting field and the "in" field.

The correct prescription, as developed by Lehmann, Symanzik and Zimmermann, requires two normalizable states $$|alpha angle and $$|eta angle, and a normalizable solution $$f(x) of the Klein-Gordon equation $$part^2+m^2=0. With these pieces one can state a correct and useful but very weak asymptotic relation:

:$$lim_{x^0 ightarrow-infty}int mathrm{d}^3x langlealpha|f(x)overleftrightarrowpart_0varphi(x)|eta angle=sqrt Z int mathrm{d}^3x langlealpha|f(x)overleftrightarrowpart_0varphi_{mathrm{in(x)|eta angle

The second member is indeed independent of time as can be shown by deriving and remembering that both $$varphi_{mathrm{in and $$f satisfy the Klein-Gordon equation.

With appropriate changes the same steps can be followed to construct an "out" field that builds "out" states. In particular the definition of the "out" field is:

:$$varphi(x)=sqrt Z varphi_{mathrm{out(x) +int mathrm{d}^4y Delta_{mathrm{adv(x-y)j(y)

where $$Delta_{mathrm{adv(x-y) is the advanced Green's function of the Klein-Gordon operator. The weak asymptotic relation between "out" field and interacting field is:

:$$lim_{x^0 ightarrow+infty}int mathrm{d}^3x langlealpha|f(x)overleftrightarrowpart_0varphi(x)|eta angle=sqrt Z int mathrm{d}^3xlanglealpha|f(x)overleftrightarrowpart_0varphi_{mathrm{out(x)|eta angle

The reduction formula for scalars

The asymptotic relations are all what is needed to obtain the LSZ reduction formula. For future convenience we start with the matrix element:

:$$mathcal M=langle eta mathrm{out}|mathrm T varphi(y_1)ldotsvarphi(y_n)|alpha p mathrm{in} angle

which is slightly more general than an S-matrix element. Indeed, $$mathcal M is the expectation value of the time-ordered product of a number of fields $$varphi(y_1)ldotsvarphi(y_n) between an "out" state and an "in" state. The "out" state can contain anything from the vacuum to an undefined number of particles, whose momenta are summarized by the index $$eta. The "in" state contains at least a particle of momentum $$p, and possibly many others, whose momenta are summarized by the index $$alpha. If there are no fields in the time-ordered product, then $$mathcal M is obviously an S-matrix element. The particle with momentum $$p can be 'extracted' from the "in" state by use of a creation operator:

:$$mathcal M=sqrt{2omega_p} langle eta mathrm{out}
mathrm Tleft [varphi(y_1)ldotsvarphi(y_n) ight] a_{mathrm{in^dagger(mathbf p)
alpha mathrm{in} angle

With the assumption that no particle with momentum "p" is present in the "out" state, that is, we are ignoring forward scattering, we can write:

:$$mathcal M=sqrt{2omega_p} langle eta mathrm{out}
mathrm Tleft [varphi(y_1)ldotsvarphi(y_n) ight] a_{mathrm{in^dagger(mathbf p)-a_{mathrm{out^dagger(mathbf p)mathrm Tleft [varphi(y_1)ldotsvarphi(y_n) ight]
alpha mathrm{in} angle

because $$a_{mathrm{out^dagger acting on the left gives zero. Expressing the construction operators in terms of "in" and "out" fields, we have:

:$$mathcal M=-isqrt{2omega_p} int mathrm{d}^3x f_p(x)overleftrightarrowpart_0 langle eta mathrm{out}
mathrm Tleft [varphi(y_1)ldotsvarphi(y_n) ight] varphi_{mathrm{in(x)-varphi_{mathrm{out(x)mathrm Tleft [varphi(y_1)ldotsvarphi(y_n) ight]
alpha mathrm{in} angle

Now we can use the asymptotic condition to write:

:$$mathcal M=-isqrt{frac{2omega_p}{Zleft{lim_{x^0 ightarrow-infty}int mathrm{d}^3x f_p(x)overleftrightarrowpart_0langle eta mathrm{out}
mathrm Tleft [varphi(y_1)ldotsvarphi(y_n) ight] varphi(x)
alpha mathrm{in} angle- ight.:$$left. -lim_{x^0 ightarrow+infty}int mathrm{d}^3x f_p(x)overleftrightarrowpart_0langle eta mathrm{out}
varphi(x)mathrm Tleft [varphi(y_1)ldotsvarphi(y_n) ight]
alpha mathrm{in} angle ight}

Then we notice that the field $$varphi(x) can be brought inside the time-ordered product, since it appears on the right when $$x^0 ightarrow -infty and on the left when $$x^0 ightarrow +infty:

:$$mathcal M=-isqrt{frac{2omega_p}{Zleft(lim_{x^0 ightarrow-infty}-lim_{x^0 ightarrow+infty} ight)int mathrm{d}^3x f_p(x)overleftrightarrowpart_0langle eta mathrm{out}
mathrm Tleft [varphi(x)varphi(y_1)ldotsvarphi(y_n) ight]
alpha mathrm{in} angle

In the following, $$x dependence in the time-ordered product is what matters, so we set::$$langle eta mathrm{out}
mathrm Tleft [varphi(x)varphi(y_1)ldotsvarphi(y_n) ight]
alpha mathrm{in} angle=eta(x)

It's easy to show by explicitly carrying out the time integration that:

:$$mathcal M=isqrt{frac{2omega_p}{Zint mathrm{d}(x^0)part_0int mathrm{d}^3x f_p(x)overleftrightarrowpart_0eta(x)

so that, by explicit time derivation, we have:

:$$mathcal M=isqrt{frac{2omega_p}{Zint mathrm{d}^4 xleft{f_p(x)part_0^2eta(x)-eta(x)part_0^2 f_p(x) ight}

By its definition we see that $$f_p(x) is a solution of the Klein-Gordon equation, which can be written as:

:$$part_0^2f_p(x)=left(Delta-m^2 ight) f_p(x)

Substituting into the expression for $$mathcal M and integrating by parts, we arrive at:

:$$mathcal M=isqrt{frac{2omega_p}{Zint mathrm{d}^4 x f_p(x)left(part_0^2-Delta+m^2 ight)eta(x)

That is:

:$$mathcal M=frac{i}{(2pi)^{3/2} Z^{1/2int mathrm{d}^4 x e^{-ipcdot x} left(Box+m^2 ight)langle eta mathrm{out}
mathrm Tleft [varphi(x)varphi(y_1)ldotsvarphi(y_n) ight]
alpha mathrm{in} angle

Starting from this result, and following the same path another particle can be extracted from the "in" state, leading to the insertion of another field in the time-ordered product. A very similar routine can extract particles from the "out" state, and the two can be iterated to get vacuum both on right and on left of the time-ordered product, leading to the general formula:

:$$langle p_1,ldots,p_n mathrm{out}|q_1,ldots,q_m mathrm{in} angle=intprod_{i=1}^{m} left{ mathrm{d}^4x_i ifrac{e^{-iq_icdot x_i{(2pi)^{3/2} Z^{1/2 left(Box_{x_i}+m^2 ight) ight} imes:$$imesprod_{j=1}^{n} left{ mathrm{d}^4y_i ifrac{e^{+ip_icdot y_i{(2pi)^{3/2} Z^{1/2 left(Box_{y_i}+m^2 ight) ight}langle 0|mathrm{T} varphi(x_1)ldotsvarphi(x_m)varphi(y_1)ldotsvarphi(y_n)|0 angle

Which is the LSZ reduction formula for Klein-Gordon scalars. It gains a much better looking aspect if it is written using the Fourier transform of the correlation function:

:$$Gamma(p_1,ldots,p_n)=int prod_{i=1}^{n} left{ mathrm{d}^4x_i e^{i p_icdot x_i} ight}langle 0|mathrm{T} varphi(x_1)ldotsvarphi(x_m)|0 angle

Using the inverse transform to substitute in the LSZ reduction formula, with some effort, the following result can be obtained:

:$$langle p_1,ldots,p_n mathrm{out}|q_1,ldots,q_m mathrm{in} angle=prod_{i=1}^{m} left{ (-i)(2pi)^{-3/2} Z^{-1/2} left(p_i^2-m^2 ight) ight} imes:$$imesprod_{j=1}^{n} left{ (-i)(2pi)^{-3/2} Z^{-1/2} left(q_i^2-m^2 ight) ight}Gamma(p_1,ldots,p_n;-q_1,ldots,-q_m)

Leaving aside normalization factors, this formula asserts that S-matrix elements are the residues of the poles that arise in the Fourier transform of the correlation functions as four-moments are put on-shell.

Reduction formula for fermions

Field strength normalization

The reason of the normalization factor $$Z in the definition of "in" and "out" fields can be understood by taking that relation between the vacuum and a single particle state $$|p angle with four-moment on-shell:

:$$langle 0|varphi(x)|p angle=sqrt Z langle 0|varphi_{mathrm{in(x)|p angle +int mathrm{d}^4y Delta_{mathrm{ret(x-y) langle 0|j(y)|p angle

Remembering that both $$varphi and $$varphi_{mathrm{in are scalar fields that lorentz transform according to:

:$$varphi(x)=e^{iPcdot x}varphi(0)e^{-iPcdot x}

where $$P^mu is the four-moment operator, we can write:

:$$e^{-ipcdot x}langle 0|varphi(0)|p angle=sqrt Z e^{-ipcdot x} langle 0|varphi_{mathrm{in(0)|p angle +int mathrm{d}^4y Delta_{mathrm{ret(x-y) langle 0|j(y)|p angle

Applying the Klein-Gordon operator $$part^2+m^2 on both sides, remembering that the four-moment $$p is on-shell and that $$Delta_{mathrm{ret is the Green's function of the operator, we obtain:

:$$0=0 +int mathrm{d}^4y delta^4(x-y) langle 0|j(y)|p angle;quadLeftrightarrowquadlangle 0|j(x)|p angle=0

So we arrive to the relation:

:$$langle 0|varphi(x)|p angle=sqrt Z langle 0|varphi_{mathrm{in(x)|p angle

which accounts for the need of the factor $$Z. The "in" field is a free field, so it can only connect one-particle states with the vacuum. That is, its expectation value between the vacuum and a many-particle state is null. On the other hand, the interacting field can also connect many-particle states to the vacuum, thanks to interaction, so the expectation values on the two sides of the last equation are different, and need a normalization factor in between. The right hand side can be computed explicitly, by expanding the "in" field in creation and annihilation operators:

:$$langle 0|varphi_{mathrm{in(x)|p angle=int frac{mathrm{d}^3q}{(2pi)^{3/2}(2omega_q)^{1/2e^{-iqcdot x} langle 0|a_{mathrm{in(mathbf q)|p angle=int frac{mathrm{d}^3q}{(2pi)^{3/2e^{-iqcdot x} langle 0|a_{mathrm{in(mathbf q)a^dagger_{mathrm{in(mathbf p)|0 angle

Using the commutation relation between $$a_{mathrm{in and $$a^dagger_{mathrm{in we obtain:

:$$langle 0|varphi_{mathrm{in(x)|p angle= frac{e^{-ipcdot x{(2pi)^{3/2

leading to the relation:

:$$langle 0|varphi(0)|p angle= sqrt frac{Z}{(2pi)^3}

by which the value of $$Z may be computed, provided that one knows how to compute $$langle 0|varphi(0)|p angle.

References

* The original paper is H. Lehmann, K. Symanzik, and W. Zimmerman, "Nuovo Cimento" 1, 205 (1955).
* A pedagogical derivation of the LSZ reduction formula can be found in M.E. Peskin and D.V. Schroeder, "An Introduction to Quantum Field Theory", Addison-Wesley, Reading, Massachusetts, 1995, Section 7.2.