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| {{Redirect|LSZ|other uses|LSZ (disambiguation)}}
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| {{Quantum field theory|cTopic=Tools}}
| |
| In [[quantum field theory]], the '''LSZ reduction formula''' is a method to calculate [[S-matrix]] elements (the [[scattering amplitude]]s) from the [[time ordered|time-ordered]] [[correlation function (quantum field theory)|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]].
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| | |
| Although the LSZ reduction formula cannot handle [[bound state]]s, [[massless particle]]s and [[topological soliton]]s, it can be generalized to cover bound states, by use of [[composite field]]s 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]].
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| ==In and Out fields==
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| | |
| [[S matrix|S-matrix elements]] are amplitudes of [[Transition of state|transitions]] between ''in'' states and ''out'' states. An ''in'' state <math>|\{p\}\ \mathrm{in}\rangle</math> describes the state of a system of particles which, in a far away past, before interacting, were moving freely with definite momenta <math>\{p\}</math>, and, conversely, an ''out'' state <math>|\{p\}\ \mathrm{out}\rangle</math> describes the state of a system of particles which, long after interaction, will be moving freely with definite momenta <math>\{p\}</math>.
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| | |
| ''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:
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| :<math>
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| S_{fi}=\langle \{q\}\ \mathrm{out}| \{p\}\ \mathrm{in}\rangle
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| </math>
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| | |
| is the [[probability amplitude]] for a set of particles which were prepared with definite momenta <math>\{p\}</math> to interact and be measured later as a new set of particles with momenta <math>\{q\}</math>. | |
| | |
| 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.
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| | |
| Just to fix ideas, suppose we deal with a [[Scalar field (quantum field theory)|Klein-Gordon field]] that interacts in some way which doesn't concern us:
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| :<math>
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| \mathcal L= \frac 1 2 \part_\mu \varphi\part^\mu \varphi - \frac 1 2 m_0^2 \varphi^2 +\mathcal L_{\mathrm{int}}
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| </math>
| |
| | |
| <math>\mathcal L_{\mathrm{int}}</math> may contain a [[Nonlinear scalar field theory|self interaction]] <math>g\ \varphi^3</math> or interaction with other fields, like a [[Yukawa interaction]] <math>g\ \varphi\bar\psi\psi</math>. From this [[Lagrangian]], using [[Euler-Lagrange equation]]s, the equation of motion follows:
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| | |
| :<math>
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| \left(\part^2+m_0^2\right)\varphi(x)=j_0(x)
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| </math>
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| | |
| where, if <math>\mathcal L_{\mathrm{int}}</math> does not contain derivative couplings:
| |
| | |
| :<math>
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| j_0=\frac{\part\mathcal L_{\mathrm{int}}}{\part \varphi}
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| </math>
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| We may expect the ''in'' field to resemble the asymptotic behaviour of the interacting field as <math>x^0\rightarrow-\infty</math>, making the assumption that in the far away past interaction described by the current <math>j_0</math> is negligible, as particles are far from each other. This hypothesis is named the '[[adiabatic theorem|adiabatic hypothesis]]'. However [[self-energy|self interaction]] never fades away and, besides many other effects, it causes a difference between the Lagrangian mass <math>m_0</math> and the physical mass <math>m</math> of the <math>\varphi</math> [[boson]]. This fact must be taken into account by rewriting the equation of motion as follows:{{Citation needed|date=August 2011}}
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| :<math>
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| \left(\part^2+m^2\right)\varphi(x)=j_0(x)+\left(m^2-m_0^2\right)\varphi(x)=j(x)
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| </math>
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| This equation can be solved formally using the retarded [[Green's function]] of the Klein-Gordon operator <math>\partial^2+m^2</math>:
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| :<math>
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| \Delta_{\mathrm{ret}}(x)=i\theta\left(x^0\right)
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| \int \frac{\mathrm{d}^3k}{(2\pi)^3 2\omega_k}
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| \left(e^{-ik\cdot x}-e^{ik\cdot x}\right)_{k^0=\omega_k};\quad
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| \omega_k=\sqrt{\mathbf{k}^2+m^2}
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| </math>
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| | |
| allowing us to split interaction from asymptotic behaviour. The solution is:
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| :<math>
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| \varphi(x)=\sqrt Z \varphi_{\mathrm{in}}(x) +\int \mathrm{d}^4y \Delta_{\mathrm{ret}}(x-y)j(y)
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| </math>
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| The factor <math>\sqrt Z</math> is a normalization factor that will come handy later, the field <math>\varphi_{\mathrm{in}}</math> is a solution of the [[Homogeneous differential equation|homogeneous equation]] associated with the equation of motion:
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| | |
| :<math>
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| \left(\part^2+m^2\right) \varphi_{\mathrm{in}}(x)=0
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| </math>,
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| and hence is a [[free field]] which describes an incoming unperturbed wave, while the last term of the solution gives the [[Perturbation theory (quantum mechanics)|perturbation]] of the wave due to interaction.
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| The field <math>\varphi_{\mathrm{in}}</math> is indeed the ''in'' field we were seeking, as it describes the asymptotic behaviour of the interacting field as <math>x^0\rightarrow-\infty</math>, though this statement will be made more precise later. It is a free scalar field so it can be expanded in flat waves:
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| :<math>
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| \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\}
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| </math>
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| | |
| where:
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| :<math>
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| f_k(x)=\left.\frac{e^{-ik\cdot x}}{(2\pi)^{3/2}(2\omega_k)^{1/2}}\right|_{k^0=\omega_k}
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| </math>
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| The inverse function for the coefficients in terms of the field can be easily obtained and put in the elegant form:
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| :<math>
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| a_{\mathrm{in}}(\mathbf{k})=i\int \mathrm{d}^3x f^*_k(x)\overleftrightarrow\partial_0\varphi_{\mathrm{in}}(x)
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| </math>
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| where:
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| | |
| <math>
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| {\mathrm{g}}{\overleftrightarrow\partial}_0 f = \mathrm{g}\partial_0 f -f\partial_0 \mathrm{g}.
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| </math>
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| The [[Fourier coefficient]]s satisfy the algebra of [[creation and annihilation operators]]:
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| :<math>
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| [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});
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| </math>
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| and they can be used to build ''in'' states in the usual way:
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| :<math>
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| \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
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| </math>
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| 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:
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| | |
| :<math>
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| \varphi(x)\sim\sqrt Z\varphi_{\mathrm{in}}(x)\quad \mathrm{as}\quad x^0\rightarrow-\infty
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| </math>
| |
| | |
| implicitly making the assumption that all interactions become negligible when particles are far away from each other. Yet the current <math>j(x)</math> contains also self interactions like those producing the mass shift from <math>m_0</math> to <math>m</math>. 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.
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| The correct prescription, as developed by Lehmann, Symanzik and Zimmermann, requires two normalizable states <math>|\alpha\rangle</math> and <math>|\beta\rangle</math>, and a normalizable solution <math>f(x)</math> of the Klein-Gordon equation <math>(\part^2+m^2)f(x)=0</math>. With these pieces one can state a correct and useful but very weak asymptotic relation:
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| :<math>
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| \lim_{x^0\rightarrow-\infty}
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| \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi(x)|\beta\rangle=
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| \sqrt Z \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi_{\mathrm{in}}(x)|\beta\rangle
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| </math>
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| | |
| The second member is indeed independent of time as can be shown by deriving and remembering that both <math>\varphi_{\mathrm{in}}</math> and <math>f</math> satisfy the Klein-Gordon equation.
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| 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:
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| :<math>
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| \varphi(x)=\sqrt Z \varphi_{\mathrm{out}}(x) +\int \mathrm{d}^4y \Delta_{\mathrm{adv}}(x-y)j(y)
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| </math>
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| | |
| where <math>\Delta_{\mathrm{adv}}(x-y)</math> is the advanced Green's function of the Klein-Gordon operator. The weak asymptotic relation between ''out'' field and interacting field is:
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| | |
| :<math>
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| \lim_{x^0\rightarrow+\infty}
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| \int \mathrm{d}^3x \langle\alpha|f(x)\overleftrightarrow\part_0\varphi(x)|\beta\rangle=
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| \sqrt Z \int \mathrm{d}^3x
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| \langle\alpha|f(x)\overleftrightarrow\part_0\varphi_{\mathrm{out}}(x)|\beta\rangle
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| </math>
| |
| | |
| ==The reduction formula for scalars==
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| The asymptotic relations are all that is needed to obtain the LSZ reduction formula. For future convenience we start with the matrix element:
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| :<math>
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| \mathcal M=\langle \beta\ \mathrm{out}|\mathrm T\ \varphi(y_1)\ldots\varphi(y_n)|\alpha\ p\ \mathrm{in}\rangle
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| </math>
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| | |
| which is slightly more general than an S-matrix element. Indeed, <math>\mathcal M</math> is the expectation value of the [[Path-ordering|time-ordered product]] of a number of fields <math>\varphi(y_1)\ldots\varphi(y_n)</math> 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 <math>\beta</math>. The ''in'' state contains at least a particle of momentum <math>p</math>, and possibly many others, whose momenta are summarized by the index <math>\alpha</math>. If there are no fields in the time-ordered product, then <math>\mathcal M</math> is obviously an S-matrix element. The particle with momentum <math>p</math> can be 'extracted' from the ''in'' state by use of a creation operator:
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| :<math>
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| \mathcal M=\sqrt{2\omega_p}\
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| \langle \beta\ \mathrm{out}|
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| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
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| a_{\mathrm{in}}^\dagger(\mathbf p)
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| |\alpha\ \mathrm{in}\rangle
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| </math>
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| 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:
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| :<math>
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| \mathcal M=\sqrt{2\omega_p}\
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| \langle \beta\ \mathrm{out}|
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| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
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| a_{\mathrm{in}}^\dagger(\mathbf p)-
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| a_{\mathrm{out}}^\dagger(\mathbf p)
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| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
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| |\alpha\ \mathrm{in}\rangle
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| </math>
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| | |
| because <math>a_{\mathrm{out}}^\dagger</math> acting on the left gives zero. Expressing the construction operators in terms of ''in'' and ''out'' fields, we have:
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| | |
| :<math>
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| \mathcal M=-i\sqrt{2\omega_p}\
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| \int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
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| \langle \beta\ \mathrm{out}|
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| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
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| \varphi_{\mathrm{in}}(x)-
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| \varphi_{\mathrm{out}}(x)
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| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
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| |\alpha\ \mathrm{in}\rangle
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| </math>
| |
| | |
| Now we can use the asymptotic condition to write:
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| | |
| :<math>
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| \mathcal M=-i\sqrt{\frac{2\omega_p}{Z}}\left\{
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| \lim_{x^0\rightarrow-\infty}
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| \int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
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| \langle \beta\ \mathrm{out}|
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| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
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| \varphi(x)
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| |\alpha\ \mathrm{in}\rangle-
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| \right.
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| </math>
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| :<math>
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| \left. -\lim_{x^0\rightarrow+\infty}
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| \int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
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| \langle \beta\ \mathrm{out}|
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| \varphi(x)
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| \mathrm T\left[\varphi(y_1)\ldots\varphi(y_n)\right]
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| |\alpha\ \mathrm{in}\rangle
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| \right\}
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| </math>
| |
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| Then we notice that the field <math>\varphi(x)</math> can be brought inside the time-ordered product, since it appears on the right when <math>x^0\rightarrow -\infty</math> and on the left when <math>x^0\rightarrow +\infty</math>:
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| | |
| :<math>
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| \mathcal M=-i\sqrt{\frac{2\omega_p}{Z}}
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| \left(\lim_{x^0\rightarrow-\infty}-\lim_{x^0\rightarrow+\infty}\right)
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| \int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
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| \langle \beta\ \mathrm{out}|
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| \mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right]
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| |\alpha\ \mathrm{in}\rangle
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| </math>
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| In the following, <math>x</math> dependence in the time-ordered product is what matters, so we set:
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| :<math>
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| \langle \beta\ \mathrm{out}|
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| \mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right]
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| |\alpha\ \mathrm{in}\rangle=
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| \eta(x)
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| </math>
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| | |
| It's easy to show by explicitly carrying out the time integration that:
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| | |
| :<math>
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| \mathcal M=i\sqrt{\frac{2\omega_p}{Z}}
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| \int \mathrm{d}(x^0)\part_0
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| \int \mathrm{d}^3x f_p(x)\overleftrightarrow\part_0
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| \eta(x)
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| </math>
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| so that, by explicit time derivation, we have:
| |
| | |
| :<math>
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| \mathcal M=i\sqrt{\frac{2\omega_p}{Z}}
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| \int \mathrm{d}^4 x\left\{f_p(x)\part_0^2\eta(x)-\eta(x)\part_0^2 f_p(x)\right\}
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| </math>
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| By its definition we see that <math>f_p(x)</math> is a solution of the Klein-Gordon equation, which can be written as:
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| | |
| :<math>
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| \part_0^2f_p(x)=\left(\Delta-m^2\right) f_p(x)
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| </math>
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| Substituting into the expression for <math>\mathcal M</math> and integrating by parts, we arrive at:
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| | |
| :<math>
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| \mathcal M=i\sqrt{\frac{2\omega_p}{Z}}
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| \int \mathrm{d}^4 x f_p(x)\left(\part_0^2-\Delta+m^2\right)\eta(x)
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| </math>
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| That is:
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| :<math>
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| \mathcal M=\frac{i}{(2\pi)^{3/2} Z^{1/2}}
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| \int \mathrm{d}^4 x e^{-ip\cdot x} \left(\Box+m^2\right)
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| \langle \beta\ \mathrm{out}|
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| \mathrm T\left[\varphi(x)\varphi(y_1)\ldots\varphi(y_n)\right]
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| |\alpha\ \mathrm{in}\rangle
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| </math>
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| 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:
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| | |
| :<math> | |
| \langle p_1,\ldots,p_n\ \mathrm{out}|q_1,\ldots,q_m\ \mathrm{in}\rangle=\int
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| \prod_{i=1}^{m}
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| \left\{
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| \mathrm{d}^4x_i\
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| i\frac{e^{-iq_i\cdot x_i}}{(2\pi)^{3/2} Z^{1/2}}
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| \left(\Box_{x_i}+m^2\right)
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| \right\}\times
| |
| </math>
| |
| :<math>
| |
| \times
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| \prod_{j=1}^{n}
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| \left\{
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| \mathrm{d}^4y_j\
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| i\frac{e^{+ip_j\cdot y_j}}{(2\pi)^{3/2} Z^{1/2}}
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| \left(\Box_{y_j}+m^2\right)
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| \right\}
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| \langle 0|\mathrm{T}\ \varphi(x_1)\ldots\varphi(x_m)\varphi(y_1)\ldots\varphi(y_n)|0\rangle
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| </math>
| |
| | |
| 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:
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| | |
| :<math>
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| \Gamma(p_1,\ldots,p_n)=\int
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| \prod_{i=1}^{n}
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| \left\{
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| \mathrm{d}^4x_i\
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| e^{i p_i\cdot x_i}
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| \right\}
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| \langle 0|\mathrm{T}\ \varphi(x_1)\ldots\varphi(x_n)|0\rangle
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| </math>
| |
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| Using the inverse transform to substitute in the LSZ reduction formula, with some effort, the following result can be obtained:
| |
| | |
| :<math>
| |
| \langle p_1,\ldots,p_n\ \mathrm{out}|q_1,\ldots,q_m\ \mathrm{in}\rangle=
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| \prod_{i=1}^{m}
| |
| \left\{
| |
| (-i)(2\pi)^{-3/2} Z^{-1/2}
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| \left(p_i^2-m^2\right)
| |
| \right\}\times
| |
| </math>
| |
| :<math>
| |
| \times
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| \prod_{j=1}^{n}
| |
| \left\{
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| (-i)(2\pi)^{-3/2} Z^{-1/2}
| |
| \left(q_j^2-m^2\right)
| |
| \right\}
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| \Gamma(p_1,\ldots,p_n;-q_1,\ldots,-q_m)
| |
| </math>
| |
| | |
| 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.
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| | |
| ==Reduction formula for fermions==
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| {{Empty section|date=July 2010}}
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| ==Field strength normalization==
| |
| The reason of the normalization factor <math>Z</math> in the definition of ''in'' and ''out'' fields can be understood by taking that relation between the vacuum and a single particle state <math>|p\rangle</math> with four-moment on-shell:
| |
| | |
| :<math>
| |
| \langle 0|\varphi(x)|p\rangle=
| |
| \sqrt Z \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle +
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| \int \mathrm{d}^4y \Delta_{\mathrm{ret}}(x-y)
| |
| \langle 0|j(y)|p\rangle
| |
| </math>
| |
| | |
| Remembering that both <math>\varphi</math> and <math>\varphi_{\mathrm{in}}</math> are scalar fields with their lorentz transform according to:
| |
| | |
| :<math>
| |
| \varphi(x)=e^{iP\cdot x}\varphi(0)e^{-iP\cdot x}
| |
| </math>
| |
| | |
| where <math>P^\mu</math> is the four-moment operator, we can write:
| |
| | |
| :<math>
| |
| 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
| |
| </math>
| |
| | |
| Applying the Klein-Gordon operator <math>\part^2+m^2</math> on both sides, remembering that the four-moment <math>p</math> is on-shell and that <math>\Delta_{\mathrm{ret}}</math> is the Green's function of the operator, we obtain:
| |
| | |
| :<math>
| |
| 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
| |
| </math>
| |
| | |
| So we arrive to the relation:
| |
| | |
| :<math>
| |
| \langle 0|\varphi(x)|p\rangle=
| |
| \sqrt Z \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle
| |
| </math>
| |
| | |
| which accounts for the need of the factor <math>Z</math>. 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:
| |
| | |
| :<math>
| |
| \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle=
| |
| \int \frac{\mathrm{d}^3q}{(2\pi)^{3/2}(2\omega_q)^{1/2}}
| |
| e^{-iq\cdot x} \langle 0|a_{\mathrm{in}}(\mathbf q)|p\rangle=
| |
| \int \frac{\mathrm{d}^3q}{(2\pi)^{3/2}}
| |
| e^{-iq\cdot x}
| |
| \langle 0|a_{\mathrm{in}}(\mathbf q)a^\dagger_{\mathrm{in}}(\mathbf p)|0\rangle
| |
| </math>
| |
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| Using the commutation relation between <math>a_{\mathrm{in}}</math> and <math>a^\dagger_{\mathrm{in}}</math> we obtain:
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| :<math> | |
| \langle 0|\varphi_{\mathrm{in}}(x)|p\rangle= \frac{e^{-ip\cdot x}}{(2\pi)^{3/2}}
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| </math>
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| leading to the relation:
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| :<math>
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| \langle 0|\varphi(0)|p\rangle= \sqrt \frac{Z}{(2\pi)^3}
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| </math>
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| by which the value of <math>Z</math> may be computed, provided that one knows how to compute <math>\langle 0|\varphi(0)|p\rangle</math>.
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| ==See also==
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| {{Portal|Mathematics}}
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| ==References==
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| * The original paper is H. Lehmann, K. Symanzik, and W. Zimmerman, ''Nuovo Cimento'' '''1''', 205 (1955).
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| * 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.
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| {{DEFAULTSORT:Lsz Reduction Formula}}
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| [[Category:Quantum field theory]]
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