advanced_tools:non-perturbative_qft
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advanced_tools:non-perturbative_qft [2017/12/17 16:29] jakobadmin created |
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | Calculation methods beyond the usual [[theories:quantum_theory:quantum_field_theory:perturbation_theory|perturbation expansion]] are especially important in QCD, because the coupling becomes large at large distances and therefore the expansion is useless. | + | Calculation methods beyond the usual [[advanced_notions:quantum_field_theory:perturbation_theory|perturbation expansion]] are especially important in QCD, because the coupling becomes large at large distances and therefore the expansion is useless. |
| <blockquote> | <blockquote> | ||
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| However, in contrast to electromagnetism, the classical field equations of general Yang-Mills theory are highly complicated. While the Maxwell equations are linear and can be easily solved (the solutions describe electromagnetic waves), the analogous Yang-Mills equations are non-linear and complicated to solve. | However, in contrast to electromagnetism, the classical field equations of general Yang-Mills theory are highly complicated. While the Maxwell equations are linear and can be easily solved (the solutions describe electromagnetic waves), the analogous Yang-Mills equations are non-linear and complicated to solve. | ||
| - | This complicated structure gives rise to many interesting phenomena that can not be described by a perturbative approach. The most important example is the structure of the [[theories:quantum_theory:quantum_field_theory:qcd_vacuum|QCD vacuum]]. | + | This complicated structure gives rise to many interesting phenomena that can not be described by a perturbative approach. The most important example is the structure of the [[advanced_notions:quantum_field_theory:qcd_vacuum|QCD vacuum]]. |
| - | ** Non-perturbative methods are also approximation methods**, but a different kind of approximation. Instead small perturbations, described by an expansion in the gauge coupling, one considers semi-classical approximations. Non-perturbative methods are important to describe scenarios where the usual perturbation theory is not applicable. A good example are tunneling processes. For example, [[the_standard_model:qcd_vacuum|QCD vacuum]] posses an infinite number of ground states with equal energy. With perturbation theory we can only describe small perturbations around such a minimum. However in a quantum theory, there can be tunneling processes between minimums of a potential. Such processes are called [[quantum_field_theory:notions:instantons|instantons]]. A perturbative approach would never notice anything about the other minimas and is therefore not able to describe the true ground state, which is a superpositon of all "classical" ground states, with tunneling processes connecting them. At first it is confusing why such processes are described by a semi-classical approximation, because there is no tunneling in classical physics. However, the semi-classical approach is merely a trick to identify the dominant contributions to the path integral. We focus on these dominant contributions, because we can not compute the path integral exactly. We call the saddle points of the action classical paths and these dominante the sum over all possible paths. Another trick to make this idea work, is to redefine the time as $i t$, i.e. make the time complex. This is necessary, because there is no tunneling in classical physics. By making the time imaginary, in some sense, we flip the potential upside down, and thus classical paths become possible. Another way to see why imaginary times are useful to describe tunneling processes, is to remember the usual quantum mechanical situation of tunneling through a potential barrier. Outside of the barrier, the wave function oscillated $\mathrm{e} {i\omega t}$. However, in the potential the wave function is damped exponentially:$\mathrm{e} {- \omega t}$. The connection between a "usual" wave function, and a tunneling wave function is therefore $t \to i t$. These points are described in the following sections in more detail. | + | ** Non-perturbative methods are also approximation methods**, but a different kind of approximation. Instead small perturbations, described by an expansion in the gauge coupling, one considers semi-classical approximations. Non-perturbative methods are important to describe scenarios where the usual perturbation theory is not applicable. A good example are tunneling processes. For example, the [[advanced_notions:quantum_field_theory:qcd_vacuum|QCD vacuum]] posses an infinite number of ground states with equal energy. With perturbation theory we can only describe small perturbations around such a minimum. However in a quantum theory, there can be tunneling processes between minimums of a potential. Such processes are called [[advanced_notions:quantum_field_theory:instantons|instantons]]. A perturbative approach would never notice anything about the other minimas and is therefore not able to describe the true ground state, which is a superpositon of all "classical" ground states, with tunneling processes connecting them. At first it is confusing why such processes are described by a semi-classical approximation, because there is no tunneling in classical physics. However, the semi-classical approach is merely a trick to identify the dominant contributions to the path integral. We focus on these dominant contributions, because we can not compute the path integral exactly. We call the saddle points of the action classical paths and these dominante the sum over all possible paths. Another trick to make this idea work, is to redefine the time as $i t$, i.e. make the time complex. This is necessary, because there is no tunneling in classical physics. By making the time imaginary, in some sense, we flip the potential upside down, and thus classical paths become possible. Another way to see why imaginary times are useful to describe tunneling processes, is to remember the usual quantum mechanical situation of tunneling through a potential barrier. Outside of the barrier, the wave function oscillated $\mathrm{e}^{i\omega t}$. However, in the potential the wave function is damped exponentially:$\mathrm{e}^{- \omega t}$. The connection between a "usual" wave function, and a tunneling wave function is therefore $t \to i t$. These points are described in the following sections in more detail. |
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| <tabbox Student> | <tabbox Student> | ||
| + | * Advanced Topics in Quantum Field Theory by M. Shifman | ||
| - | <note tip> | ||
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | ||
| - | </note> | ||
| - | |||
| <tabbox Researcher> | <tabbox Researcher> | ||
| - | |||
| - | <note tip> | ||
| - | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| - | </note> | ||
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| Usually in QFT we use perturbation theory. However, there are phenomena that can not be described by this method. A good example is the double well potential $V(q)=\frac{\lambda}{4!}(q^2-a^2)^2$: | Usually in QFT we use perturbation theory. However, there are phenomena that can not be described by this method. A good example is the double well potential $V(q)=\frac{\lambda}{4!}(q^2-a^2)^2$: | ||
| - | [{{ :advanced_tools:non-perturbative_qft:dobulewell.jpg?nolink |Source: http://physics.stackexchange.com/questions/131272/is-quantum-tunneling-related-to-imaginary-time}}] | + | |
| + | |||
| + | [{{ :advanced_tools:dobulewell.jpg?nolink |Source: http://physics.stackexchange.com/questions/131272/is-quantum-tunneling-related-to-imaginary-time}}] | ||
| There are two classical ground states. However, in the usual perturbation approach, we ignore this fact and expand $V$ around one of these minimas. The potential then looks exactly as the potential of the harmonic oscillator about that minimum plus anharmonic terms (cubic, quartic). The perturbation approach then yields possible wave functions and energies. However, we never see any evidence of the second minimum. Of course, it makes no difference which minimum we choose. Equally, we could expand around the other minimum. The energy levels are exactly the same in all orders of perturbation theory. However through perturbative effects there is only one true ground state and not two degenerate ground states which is suggested by perturbation theory. This shows that perturbation theory fails to describe all possible phenomena. (Source: http://www.weizmann.ac.il/particle/perez/Courses/QMII16/TA4.pdf) | There are two classical ground states. However, in the usual perturbation approach, we ignore this fact and expand $V$ around one of these minimas. The potential then looks exactly as the potential of the harmonic oscillator about that minimum plus anharmonic terms (cubic, quartic). The perturbation approach then yields possible wave functions and energies. However, we never see any evidence of the second minimum. Of course, it makes no difference which minimum we choose. Equally, we could expand around the other minimum. The energy levels are exactly the same in all orders of perturbation theory. However through perturbative effects there is only one true ground state and not two degenerate ground states which is suggested by perturbation theory. This shows that perturbation theory fails to describe all possible phenomena. (Source: http://www.weizmann.ac.il/particle/perez/Courses/QMII16/TA4.pdf) | ||
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| --> Why do we use a imaginary time?# | --> Why do we use a imaginary time?# | ||
| - | [{{ :advanced_tools:non-perturbative_qft:potentialflip2.png?nolink |Source: http://www.pks.mpg.de/~topo14/slides/Niemi_text.pdf}}] | + | [{{ :advanced_tools:potentialflip2.png?nolink |Source: http://www.pks.mpg.de/~topo14/slides/Niemi_text.pdf}}] |
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| Consider a particle with energy $E$ sitting in the minimum of a double well potential, for example, at B in the following picture: | Consider a particle with energy $E$ sitting in the minimum of a double well potential, for example, at B in the following picture: | ||
| - | [{{ :advanced_tools:non-perturbative_qft:dobulewell.jpg?nolink |Source: http://physics.stackexchange.com/questions/131272/is-quantum-tunneling-related-to-imaginary-time}}] | + | [{{ :advanced_tools:dobulewell.jpg?nolink |Source: http://physics.stackexchange.com/questions/131272/is-quantum-tunneling-related-to-imaginary-time}}] |
advanced_tools/non-perturbative_qft.1513524540.txt.gz · Last modified: 2017/12/17 15:29 (external edit)
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