LSZ reduction formula

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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[edit]

S-matrix elements are amplitudes of transitions between in states and out states. An in state |\{p\}\ \mathrm{in}\rangle 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}\rangle 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}\rangle

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 concern us:

\mathcal L= \frac 1 2 \part_\mu \varphi\part^\mu \varphi - \frac 1 2 m_0^2 \varphi^2 +\mathcal L_{\mathrm{int}}

\mathcal L_{\mathrm{int}} may contain a self interaction 3 or interaction with other fields, like a Yukawa interaction g\ \varphi\bar\psi\psi. From this Lagrangian, using Euler–Lagrange equations, the equation of motion follows:

\left(\part^2+m_0^2\right)\varphi(x)=j_0(x)

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

j_0=\frac{\part\mathcal L_{\mathrm{int}}}{\part \varphi}

We may expect the in field to resemble the asymptotic behaviour of the interacting field as x0 → −∞, making the assumption that in the far away past interaction described by the current j0 is negligible, as particles are far from each other. This hypothesis is named the adiabatic hypothesis. However self interaction never fades away and, besides many other effects, it causes a difference between the Lagrangian mass m0 and the physical mass m of the φ boson. This fact must be taken into account by rewriting the equation of motion as follows:[citation needed]

\left(\part^2+m^2\right)\varphi(x)=j_0(x)+\left(m^2-m_0^2\right)\varphi(x)=j(x)

This equation can be solved formally using the retarded Green's function of the Klein–Gordon operator \partial^2+m^2:

\Delta_{\mathrm{ret}}(x)=i\theta\left(x^0\right)\int \frac{\mathrm{d}^3k}{(2\pi)^3 2\omega_k} \left(e^{-ik\cdot x}-e^{ik\cdot x}\right)_{k^0=\omega_k}\qquad \omega_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 Z is a normalization factor that will come handy later, the field φin is a solution of the homogeneous equation associated with the equation of motion:

\left(\part^2+m^2\right) \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 φin is indeed the in field we were seeking, as it describes the asymptotic behaviour of the interacting field as x0 → −∞, 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})\right\}

where:

f_k(x)=\left.\frac{e^{-ik\cdot x}}{(2\pi)^{\frac{3}{2}}(2\omega_k)^{\frac{1}{2}}}\right|_{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})=i\int \mathrm{d}^3x f^*_k(x)\overleftrightarrow\partial_0\varphi_{\mathrm{in}}(x)

where:

{\mathrm{g}}{\overleftrightarrow\partial}_0 f = \mathrm{g}\partial_0 f -f\partial_0 \mathrm{g}.

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}\right\rangle=\sqrt{2\omega_{k_1}}a_{\mathrm{in}}^\dagger(\mathbf{k}_1)\ldots \sqrt{2\omega_{k_n}}a_{\mathrm{in}}^\dagger(\mathbf{k}_n)|0\rangle

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)\sim\sqrt Z\varphi_{\mathrm{in}}(x)\qquad \mathrm{as}\quad x^0\to-\infty

implicitly making the assumption that all interactions become negligible when particles are far away from each other. Yet the current j(x) contains also self interactions like those producing the mass shift from m0 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\rangle and |\beta\rangle, and a normalizable solution &nbspf (x) of the Klein–Gordon equation (\part^2+m^2)f(x)=0. With these pieces one can state a correct and useful but very weak asymptotic relation:

\lim_{x^0\to-\infty} \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi(x)|\beta\rangle= \sqrt Z \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi_{\mathrm{in}}(x)|\beta\rangle

The second member is indeed independent of time as can be shown by deriving and remembering that both φin 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 Δadv(xy) 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\to \infty} \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi(x)|\beta\rangle= \sqrt Z \int \mathrm{d}^3x
\langle\alpha|f(x)\overleftrightarrow\part_0\varphi_{\mathrm{out}}(x)|\beta\rangle

The reduction formula for scalars[edit]

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

\mathcal M=\langle \beta\ \mathrm{out}|\mathrm{T}\varphi(y_1)\ldots\varphi(y_n)|\alpha\ p\ \mathrm{in}\rangle

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)\cdots\varphi(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 β. The in state contains at least a particle of momentum p, and possibly many others, whose momenta are summarized by the index α. 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{2\omega_p}\ \left \langle \beta\ \mathrm{out} \bigg| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right] a_{\mathrm{in}}^\dagger(\mathbf p) \bigg|\alpha\ \mathrm{in} \right \rangle

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{2\omega_p}\ \left \langle \beta\ \mathrm{out} \bigg| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right] a_{\mathrm{in}}^\dagger (\mathbf p)- a_{\mathrm{out}}^\dagger(\mathbf p) \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right] \bigg|\alpha\ \mathrm{in} \right \rangle

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=-i\sqrt{2\omega_p}\ \int \mathrm{d}^3x f_p(x)\overleftrightarrow{\part_0} \left\langle \beta\ \mathrm{out} \bigg| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right] \varphi_{\mathrm{in}}(x)- \varphi_{\mathrm{out}}(x) \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right] \bigg|\alpha\ \mathrm{in}\right \rangle

Now we can use the asymptotic condition to write:

\mathcal M= -i\sqrt{\frac{2\omega_p}{Z}} \left\{ \lim_{x^0\to-\infty} \int \mathrm{d}^3x f_p(x)\overleftrightarrow{\part_0} \langle \beta\ \mathrm{out}| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right] \varphi(x) |\alpha\ \mathrm{in}\rangle-\lim_{x^0\to\infty} \int \mathrm{d}^3x f_p(x)\overleftrightarrow{\part_0} \langle \beta\ \mathrm{out}| \varphi(x) \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right] |\alpha\ \mathrm{in}\rangle \right\}

Then we notice that the field φ(x) can be brought inside the time-ordered product, since it appears on the right when x0 → −∞ and on the left when x0 → ∞:

\mathcal M=-i\sqrt{\frac{2\omega_p}{Z}} \left(\lim_{x^0\to-\infty}-\lim_{x^0\to \infty}\right) \int \mathrm{d}^3x f_p(x) \overleftrightarrow{\part_0} \langle \beta\ \mathrm{out}| \mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right] |\alpha\ \mathrm{in} \rangle

In the following, x dependence in the time-ordered product is what matters, so we set:

\langle \beta\ \mathrm{out}| \mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right] |\alpha\ \mathrm{in}\rangle= \eta(x)

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

 \mathcal M=i\sqrt{\frac{2\omega_p}{Z}} \int \mathrm{d}(x^0)\part_0 \int \mathrm{d}^3x f_p(x)\overleftrightarrow{\part_0}\eta(x)

so that, by explicit time derivation, we have:

\mathcal M=i\sqrt{\frac{2\omega_p}{Z}} \int \mathrm{d}^4 x\left\{f_p(x)\part_0^2\eta(x)-\eta(x)\part_0^2 f_p(x)\right\}

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

\part_0^2f_p(x)=\left(\Delta-m^2\right) f_p(x)

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

\mathcal M=i\sqrt{\frac{2\omega_p}{Z}} \int \mathrm{d}^4 x f_p(x)\left(\part_0^2-\Delta+m^2\right)\eta(x)

That is:

 \mathcal M=\frac{i}{(2\pi)^{\frac{3}{2}} Z^{\frac{1}{2}}} \int \mathrm{d}^4 x e^{-ip\cdot x} \left(\Box+m^2\right)\langle \beta\ \mathrm{out}| \mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right] |\alpha\ \mathrm{in}\rangle

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}\rangle=\int \prod_{i=1}^{m} \left\{\mathrm{d}^4x_i \frac{i e^{-iq_i\cdot x_i} \left(\Box_{x_i}+m^2\right)}{(2\pi)^{\frac{3}{2}} Z^{\frac{1}{2}}}  \right\} \prod_{j=1}^{n} \left\{ \mathrm{d}^4y_j \frac{i e^{ip_j\cdot y_j}\left(\Box_{y_j}+m^2\right)}{(2\pi)^{\frac{3}{2}} Z^{\frac{1}{2}}}  \right\} \langle 0|\mathrm{T} \varphi(x_1)\ldots\varphi(x_m)\varphi(y_1)\ldots\varphi(y_n)|0\rangle

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 \left (p_1,\ldots,p_n \right )=\int \prod_{i=1}^{n} \left\{\mathrm{d}^4x_i e^{i p_i\cdot x_i} \right\} \langle 0|\mathrm{T}\ \varphi(x_1)\ldots\varphi(x_n)|0\rangle

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}\rangle= \prod_{i=1}^{m} \left\{-\frac{i\left(p_i^2-m^2\right)}{(2\pi)^{\frac{3}{2}} Z^{\frac{1}{2}}} \right\} \prod_{j=1}^{n} \left\{ -\frac{i\left(q_j^2-m^2\right)}{(2\pi)^{\frac{3}{2}} Z^{\frac{1}{2}}}  \right\} \Gamma \left (p_1,\ldots,p_n;-q_1,\ldots,-q_m \right )

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-momenta are put on-shell.

Reduction formula for fermions[edit]

Field strength normalization[edit]

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\rangle with four-moment on-shell:

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

Remembering that both φ and φin are scalar fields with their lorentz transform according to:

\varphi(x)=e^{iP\cdot x}\varphi(0)e^{-iP\cdot x}

where Pμ is the four-moment operator, we can write:

 e^{-ip\cdot x}\langle 0|\varphi(0)|p\rangle= \sqrt Z e^{-ip\cdot x} \langle 0|\varphi_{\mathrm{in}}(0)|p\rangle + \int \mathrm{d}^4y \Delta_{\mathrm{ret}}(x-y)\langle 0|j(y)|p\rangle

Applying the Klein–Gordon operator 2 + m2 on both sides, remembering that the four-moment p is on-shell and that Δret 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\rangle; \quad\Leftrightarrow\quad \langle 0|j(x)|p\rangle=0

So we arrive to the relation:

\langle 0|\varphi(x)|p\rangle= \sqrt Z \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle

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\rangle= \int \frac{\mathrm{d}^3q}{(2\pi)^{\frac{3}{2}}(2\omega_q)^{\frac{1}{2}}} e^{-iq\cdot x} \langle 0|a_{\mathrm{in}}(\mathbf q)|p\rangle= \int \frac{\mathrm{d}^3q}{(2\pi)^{\frac{3}{2}}} e^{-iq\cdot x} \langle 0|a_{\mathrm{in}}(\mathbf q)a^\dagger_{\mathrm{in}}(\mathbf p)|0\rangle

Using the commutation relation between ain and a^\dagger_{\mathrm{in}} we obtain:

 \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle= \frac{e^{-ip\cdot x}}{(2\pi)^{\frac{3}{2}}}

leading to the relation:

\langle 0|\varphi(0)|p\rangle= \sqrt \frac{Z}{(2\pi)^3}

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

See also[edit]

References[edit]

  • 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.