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If we stretch the definition of a Feynman diagram, then yes: the technique can be applied to any problem where you use perturbation theory. But if by Feynman diagram you mean the exact same philosophy behind QFT, then in principle the answer is no: it only works in those problems where you have the same algebraic structure of QFT.

In QFT, there are two approaches to Feynman diagrams: canonical quantisation and path integrals. The latter can be summarised in the gaussian-type integrals $$ g_n\equiv\int_{-\infty}^{+\infty}x^{2n}\mathrm e^{-\frac{1}{2}ax^2}=\sqrt{\frac{2\pi}{a}}a^{-n}(2n-1)!! \tag{1} $$

Feynman diagrams arises when we try to compute $g_n$ using combinatorial arguments together with the generating integral $$ z_j=\int_{-\infty}^{+\infty} \mathrm e^{-\frac{1}{2}ax^2+jx} \tag{2} $$ where, for example, $g_2=[\partial^2z_j]_{j=0}$. For more details, see Mathematical Ideas and Notions of Quantum Field Theory.

This means: whenever you can invoke combinatorics to some problem at hand, then in principle you can use diagrams to represent the different combinations, which in turns means you can restate the problem through Feynman diagrams.

Canonical cuantisation, on the other hand, deals with operators with the algebraic structure $$ [a_i,a_j]=0\qquad [a_i,a_j^\dagger]\sim \delta_{ij} \tag{3} $$

In this case, you can use this algebraic structure to prove Wick's theorem, which is (almost) the same as Feynman diagrams. This means: if you have any theory on some Hilbert space, where the operators satisfy $(3)$ (e.g., some Sturm-Liouville problem), then in principle you can use Feynman diagrams to solve problems.

The trivial example is, of course, the differential operator $a:L^2(\mathbb R)\to L^2(\mathbb R)$ $$ a f(x)=\left(x+\frac{\mathrm d}{\mathrm dx}\right)f(x) \tag{4} $$ i.e., the ladder operator of the quantum harmonic oscillator. You can use Feynmal diagrams to calculate any "expectation value" $$ \int_{-\infty}^{+\infty} p(x) f(x)\mathrm dx \tag{5} $$ where $p(x)$ is any polynomial. As the eigenfunctions of $a^\dagger a$ are (Hermite) polynomials times a gaussian exponential, we get $(1)$ back.

To sum up: if we extend the notion of Feynman diagrams, then I guess you could use them for most problems (though I'm not sure the usefulness of this). On the other hand, the standard meaning of Feynman diagrams can only be used if you are doing the same thing they were invented for, that is, path integrals, or something that shares the same algebraic structure.

A nice example of the use of Feynman diagrams to general problems is given in Solving Classical Field Equations, where the author explains how diagrams can be used to solve non-linear PDE's perturbatively. In this sense, one could say many problems in physics can be solved using Feynman diagrams, because of the ubiquity of differential equations in physics.

Another example of the use of Feynman diagrams for perturbation theory is given in this nice post by QMechanic, where you can see that Feynman diagrams can be used in (non-relativistic) quantum mechanics to simplify the evaluation of higher-order terms in perturbation theory.

If we stretch the definition of a Feynman diagram, then yes: the technique can be applied to any problem where you use perturbation theory. But if by Feynman diagram you mean the exact same philosophy behind QFT, then in principle the answer is no: it only works in those problems where you have the same algebraic structure of QFT.

In QFT, there are two approaches to Feynman diagrams: canonical quantisation and path integrals. The latter can be summarised in the gaussian-type integrals $$ g_n\equiv\int_{-\infty}^{+\infty}x^{2n}\mathrm e^{-\frac{1}{2}ax^2}=\sqrt{\frac{2\pi}{a}}a^{-n}(2n-1)!! \tag{1} $$

Feynman diagrams arises when we try to compute $g_n$ using combinatorial arguments together with the generating integral $$ z_j=\int_{-\infty}^{+\infty} \mathrm e^{-\frac{1}{2}ax^2+jx} \tag{2} $$ where, for example, $g_2=[\partial^2z_j]_{j=0}$. For more details, see Mathematical Ideas and Notions of Quantum Field Theory.

This means: whenever you can invoke combinatorics to some problem at hand, then in principle you can use diagrams to represent the different combinations, which in turns means you can restate the problem through Feynman diagrams.

Canonical cuantisation, on the other hand, deals with operators with the algebraic structure $$ [a_i,a_j]=0\qquad [a_i,a_j^\dagger]\sim \delta_{ij} \tag{3} $$

In this case, you can use this algebraic structure to prove Wick's theorem, which is (almost) the same as Feynman diagrams. This means: if you have any theory on some Hilbert space, where the operators satisfy $(3)$ (e.g., some Sturm-Liouville problem), then in principle you can use Feynman diagrams to solve problems.

The trivial example is, of course, the differential operator $a:L^2(\mathbb R)\to L^2(\mathbb R)$ $$ a f(x)=\left(x+\frac{\mathrm d}{\mathrm dx}\right)f(x) \tag{4} $$ i.e., the ladder operator of the quantum harmonic oscillator. You can use Feynmal diagrams to calculate any "expectation value" $$ \int_{-\infty}^{+\infty} p(x) f(x)\mathrm dx \tag{5} $$ where $p(x)$ is any polynomial. As the eigenfunctions of $a^\dagger a$ are (Hermite) polynomials times a gaussian exponential, we get $(1)$ back.

To sum up: if we extend the notion of Feynman diagrams, then I guess you could use them for most problems (though I'm not sure the usefulness of this). On the other hand, the standard meaning of Feynman diagrams can only be used if you are doing the same thing they were invented for, that is, path integrals, or something that shares the same algebraic structure.

A nice example of the use of Feynman diagrams to general problems is given in Solving Classical Field Equations, where the author explains how diagrams can be used to solve non-linear PDE's perturbatively. In this sense, one could say many problems in physics can be solved using Feynman diagrams, because of the ubiquity of differential equations in physics.

Another example of the use of Feynman diagrams for perturbation theory is given in this nice post by QMechanic, where you can see that Feynman diagrams can be used in (non-relativistic) quantum mechanics to simplify the evaluation of higher-order terms in perturbation theory.

If we stretch the definition of a Feynman diagram, then yes: the technique can be applied to any problem where you use perturbation theory. But if by Feynman diagram you mean the exact same philosophy behind QFT, then in principle the answer is no: it only works in those problems where you have the same algebraic structure of QFT.

In QFT, there are two approaches to Feynman diagrams: canonical quantisation and path integrals. The latter can be summarised in the gaussian-type integrals $$ g_n\equiv\int_{-\infty}^{+\infty}x^{2n}\mathrm e^{-\frac{1}{2}ax^2}=\sqrt{\frac{2\pi}{a}}a^{-n}(2n-1)!! \tag{1} $$

Feynman diagrams arises when we try to compute $g_n$ using combinatorial arguments together with the generating integral $$ z_j=\int_{-\infty}^{+\infty} \mathrm e^{-\frac{1}{2}ax^2+jx} \tag{2} $$ where, for example, $g_2=[\partial^2z_j]_{j=0}$. For more details, see Mathematical Ideas and Notions of Quantum Field Theory.

This means: whenever you can invoke combinatorics to some problem at hand, then in principle you can use diagrams to represent the different combinations, which in turns means you can restate the problem through Feynman diagrams.

Canonical cuantisation, on the other hand, deals with operators with the algebraic structure $$ [a_i,a_j]=0\qquad [a_i,a_j^\dagger]\sim \delta_{ij} \tag{3} $$

In this case, you can use this algebraic structure to prove Wick's theorem, which is (almost) the same as Feynman diagrams. This means: if you have any theory on some Hilbert space, where the operators satisfy $(3)$ (e.g., some Sturm-Liouville problem), then in principle you can use Feynman diagrams to solve problems.

The trivial example is, of course, the differential operator $a:L^2(\mathbb R)\to L^2(\mathbb R)$ $$ a f(x)=\left(x+\frac{\mathrm d}{\mathrm dx}\right)f(x) \tag{4} $$ i.e., the ladder operator of the quantum harmonic oscillator. You can use Feynmal diagrams to calculate any "expectation value" $$ \int_{-\infty}^{+\infty} p(x) f(x)\mathrm dx \tag{5} $$ where $p(x)$ is any polynomial. As the eigenfunctions of $a^\dagger a$ are (Hermite) polynomials times a gaussian exponential, we get $(1)$ back.

To sum up: if we extend the notion of Feynman diagrams, then I guess you could use them for most problems (though I'm not sure the usefulness of this). On the other hand, the standard meaning of Feynman diagrams can only be used if you are doing the same thing they were invented for, that is, path integrals, or something that shares the same algebraic structure.

A nice example of the use of Feynman diagrams to general problems is given in Solving Classical Field Equations, where the author explains how diagrams can be used to solve non-linear PDE's perturbatively. In this sense, one could say many problems in physics can be solved using Feynman diagrams, because of the ubiquity of differential equations in physics.

If we stretch the definition of a Feynman diagram, then yes: the technique can be applied to any problem where you use perturbation theory. But if by Feynman diagram you mean the exact same philosophy behind QFT, then in principle the answer is no: it only works in those problems where you have the same algebraic structure of QFT.

In QFT, there are two approaches to Feynman diagrams: canonical quantisation and path integrals. The latter can be summarised in the gaussian-type integrals $$ g_n\equiv\int_{-\infty}^{+\infty}x^{2n}\mathrm e^{-\frac{1}{2}ax^2}=\sqrt{\frac{2\pi}{a}}a^{-n}(2n-1)!! \tag{1} $$

Feynman diagrams arises when we try to compute $g_n$ using combinatorial arguments together with the generating integral $$ z_j=\int_{-\infty}^{+\infty} \mathrm e^{-\frac{1}{2}ax^2+jx} \tag{2} $$ where, for example, $g_2=[\partial^2z_j]_{j=0}$. For more details, see Mathematical Ideas and Notions of Quantum Field Theory.

This means: whenever you can invoke combinatorics to some problem at hand, then in principle you can use diagrams to represent the different combinations, which in turns means you can restate the problem through Feynman diagrams.

Canonical cuantisation, on the other hand, deals with operators with the algebraic structure $$ [a_i,a_j]=0\qquad [a_i,a_j^\dagger]\sim \delta_{ij} \tag{3} $$

In this case, you can use this algebraic structure to prove Wick's theorem, which is (almost) the same as Feynman diagrams. This means: if you have any theory on some Hilbert space, where the operators satisfy $(3)$ (e.g., some Sturm-Liouville problem), then in principle you can use Feynman diagrams to solve problems.

The trivial example is, of course, the differential operator $a:L^2(\mathbb R)\to L^2(\mathbb R)$ $$ a f(x)=\left(x+\frac{\mathrm d}{\mathrm dx}\right)f(x) \tag{4} $$ i.e., the ladder operator of the quantum harmonic oscillator. You can use Feynmal diagrams to calculate any "expectation value" $$ \int_{-\infty}^{+\infty} p(x) f(x)\mathrm dx \tag{5} $$ where $p(x)$ is any polynomial. As the eigenfunctions of $a^\dagger a$ are (Hermite) polynomials times a gaussian exponential, we get $(1)$ back.

To sum up: if we extend the notion of Feynman diagrams, then I guess you could use them for most problems (though I'm not sure the usefulness of this). On the other hand, the standard meaning of Feynman diagrams can only be used if you are doing the same thing they were invented for, that is, path integrals, or something that shares the same algebraic structure.

A nice example of the use of Feynman diagrams to general problems is given in Solving Classical Field Equations, where the author explains how diagrams can be used to solve non-linear PDE's perturbatively. In this sense, one could say many problems in physics can be solved using Feynman diagrams, because of the ubiquity of differential equations in physics.

Another example of the use of Feynman diagrams for perturbation theory is given in this nice post by QMechanic, where you can see that Feynman diagrams can be used in (non-relativistic) quantum mechanics to simplify the evaluation of higher-order terms in perturbation theory.

If we stretch the definition of a Feynman diagram, then yes: the technique can be applied to any problem where you use perturbation theory. But if by Feynman diagram you mean the exact same philosophy behind QFT, then in principle the answer is no: it only works in those problems where you have the same algebraic structure of QFT.

In QFT, there are two approaches to Feynman diagrams: canonical quantisation and path integrals. The latter can be summarised in the gaussian-type integrals $$ g_n\equiv\int_{-\infty}^{+\infty}x^{2n}\mathrm e^{-\frac{1}{2}ax^2}=\sqrt{\frac{2\pi}{a}}a^{-n}(2n-1)!! \tag{1} $$

Feynman diagrams arises when we try to compute $g_n$ using combinatorial arguments together with the generating integral $$ z_j=\int_{-\infty}^{+\infty} \mathrm e^{-\frac{1}{2}ax^2+jx} \tag{2} $$ where, for example, $g_2=[\partial^2z_j]_{j=0}$. For more details, see Mathematical Ideas and Notions of Quantum Field Theory.

This means: whenever you can invoke combinatorics to some problem at hand, then in principle you can use diagrams to represent the different combinations, which in turns means you can restate the problem through Feynman diagrams.

Canonical cuantisation, on the other hand, deals with operators with the algebraic structure $$ [a_i,a_j]=0\qquad [a_i,a_j^\dagger]\sim \delta_{ij} \tag{3} $$

In this case, you can use this algebraic structure to prove Wick's theorem, which is (almost) the same as Feynman diagrams. This means: if you have any theory on some Hilbert space, where the operators satisfy $(3)$ (e.g., some Sturm-Liouville problem), then in principle you can use Feynman diagrams to solve problems.

The trivial example is, of course, the differential operator $a:L^2(\mathbb R)\to L^2(\mathbb R)$ $$ a f(x)=\left(x+\frac{\mathrm d}{\mathrm dx}\right)f(x) \tag{4} $$ i.e., the ladder operator of the quantum harmonic oscillator. You can use Feynmal diagrams to calculate any "expectation value" $$ \int_{-\infty}^{+\infty} p(x) f(x)\mathrm dx \tag{5} $$ where $p(x)$ is any polynomial. As the eigenfunctions of $a^\dagger a$ are (Hermite) polynomials times a gaussian exponential, we get $(1)$ back.

To sum up: if we extend the notion of Feynman diagrams, then I guess you could use them for most problems (though I'm not sure the usefulness of this). On the other hand, the standard meaning of Feynman diagrams can only be used if you are doing the same thing they were invented for, that is, path integrals, or something that shares the same algebraic structure.

If we stretch the definition of a Feynman diagram, then yes: the technique can be applied to any problem where you use perturbation theory. But if by Feynman diagram you mean the exact same philosophy behind QFT, then in principle the answer is no: it only works in those problems where you have the same algebraic structure of QFT.

In QFT, there are two approaches to Feynman diagrams: canonical quantisation and path integrals. The latter can be summarised in the gaussian-type integrals $$ g_n\equiv\int_{-\infty}^{+\infty}x^{2n}\mathrm e^{-\frac{1}{2}ax^2}=\sqrt{\frac{2\pi}{a}}a^{-n}(2n-1)!! \tag{1} $$

Feynman diagrams arises when we try to compute $g_n$ using combinatorial arguments together with the generating integral $$ z_j=\int_{-\infty}^{+\infty} \mathrm e^{-\frac{1}{2}ax^2+jx} \tag{2} $$ where, for example, $g_2=[\partial^2z_j]_{j=0}$. For more details, see Mathematical Ideas and Notions of Quantum Field Theory.

This means: whenever you can invoke combinatorics to some problem at hand, then in principle you can use diagrams to represent the different combinations, which in turns means you can restate the problem through Feynman diagrams.

Canonical cuantisation, on the other hand, deals with operators with the algebraic structure $$ [a_i,a_j]=0\qquad [a_i,a_j^\dagger]\sim \delta_{ij} \tag{3} $$

In this case, you can use this algebraic structure to prove Wick's theorem, which is (almost) the same as Feynman diagrams. This means: if you have any theory on some Hilbert space, where the operators satisfy $(3)$ (e.g., some Sturm-Liouville problem), then in principle you can use Feynman diagrams to solve problems.

The trivial example is, of course, the differential operator $a:L^2(\mathbb R)\to L^2(\mathbb R)$ $$ a f(x)=\left(x+\frac{\mathrm d}{\mathrm dx}\right)f(x) \tag{4} $$ i.e., the ladder operator of the quantum harmonic oscillator. You can use Feynmal diagrams to calculate any "expectation value" $$ \int_{-\infty}^{+\infty} p(x) f(x)\mathrm dx \tag{5} $$ where $p(x)$ is any polynomial. As the eigenfunctions of $a^\dagger a$ are (Hermite) polynomials times a gaussian exponential, we get $(1)$ back.

To sum up: if we extend the notion of Feynman diagrams, then I guess you could use them for most problems (though I'm not sure the usefulness of this). On the other hand, the standard meaning of Feynman diagrams can only be used if you are doing the same thing they were invented for, that is, path integrals, or something that shares the same algebraic structure.

A nice example of the use of Feynman diagrams to general problems is given in Solving Classical Field Equations, where the author explains how diagrams can be used to solve non-linear PDE's perturbatively. In this sense, one could say many problems in physics can be solved using Feynman diagrams, because of the ubiquity of differential equations in physics.