This book introduces QFT for a reader with no prior knowledge of the subject. It is meant to be a textbook for advanced undergraduate or beginning postgraduate. This book introduces QFT for readers with no prior knowledge of the subject. It is meant to be a textbook for advanced undergraduate or beginning postgraduate. A First Book of Quantum Field Theory. Amitabha Lahiri, Palash B. Pal. - pages. Keyword(s): INSPIRE: book | lectures | field theory.
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Here are my notes and solutions to accompany the book A First Book of Quantum Field Theory (Second Edition) by Amitabha Lahiri & P. B. Pal. As always, no. A First Book of Quantum Field Theory by Amitabha Lahiri, , available at Book Depository with free delivery worldwide. A First Book of Quantum Field Theory book. Read reviews from world's largest community for readers. This book introduces QFT for readers with no prior kn.
Quantization of Dirac fields Using ,,5, we can construct a set of sixteen linearly independent. Therefore the matrices must be at least 4 x 4. Multiply the sum by each of r, in tUTn and take the tTau.
This, along with the anticommutation relation of Eq. It is immediately clear that the specific form of the ,,-matrices is not unique. The converse of this statement happens to be true as well. If two sets of matrices "'I" and "I" satisfy Eq. But once it is assumed, it is easy to see that since in addition the matrices "'I" and "I" both satisfy the hermiticity condition of Eq. We will not use any explicit representation of the ' '-matrices since one representation is as good as any other.
The only exception will be made in 4. This is the celebrated Dirac equation. Multiplying from the left by "'10, this can be put into the form.
It is convenient and conventional to use the Islash' notation any 4-vector aJ. J I we will henceforth write. Since the 7"'S are 4 x 4 matrioes, it follows that ", x must be a column matrix with 4 entries. Since the ' '-matrices are not unique, the solution "' x will also be non-unique. If ;j; x satisfies Eq. The Dirac equation should be relativistically covariant. This fact fixes the Lorentz transformation property of ,p x. To see this explicitly, let us assume that Eq. Thus, 4. Putting these into Eq.
Thus for infinitesimal transformations, the Dirac equation is c0variant if,p transforms by the! Finite Lorentz transformations can be obtained by exponentiating Eq. The corresponding transformation rule for ,p is. Objects defined in space-time are classified as scalars, vectors, tensors, etc. Here we find a new class of objects - column vectors whose components mix among themselves according to Eq. Such objects are called spinors even if they do not satisfy the Dirac equation.
Using the form of the infinitesimal Lorentz transformation matrix from Eq.
The first term is the familiar expression for the orbital angular momentum, but there is also a second term. This shows that the solutions of Dirac equation carries some intrinsic angular momentum, or spm. Exercise 4. This is tne renson " Ii carries a Lorentz index despite not being a vector. Although Dirac set forth to avoid the negative energy solutions, the irony of the situation is that the Dirac equation also has such solutions, just like the Klein-Gordon equation.
To see that, let us assume that we are trying a plane wave solution in the rest frame of a particle. The time dependence of this. Thus we obtain the solution of the eigenvalue equation to be. Thus both positive and negative eigenvalues will occur in a single particle interpretation of the Dirac equation. For a general value of the 3-momentum p, let us represent solutions in the form.
The normalization of the spinors has not been specified yet. Let us now fix them with the conditions. DeritJe a. In particular, putting J1. Note that this is an incorrect normalization 4. This leads to. Take, for example, the first equation. Appl-y the matrices on hoth sides on the fouT' basis spinol'S and show that the T'esults are the same. If both sides act similo: So far, we have done everything in a representation-independent way.
As we remarked earlier, all representations are equivalent, and for. However, it is useful to derive the plane wave solutions in a particular representation in order to gain some insight.
To find the solutions, we put the representation of the -r-matrices into Eq. First, note that. Thus, if we write the u-spinors also as u. These are two homogeneous equations with two unknowns, and s0lutions exist since the determinant of the co-efficients is zero, which can be seen by explicit calculation. We can now use anyone of the equations in Eq. Using the second one, we get. We can now write down the solutions for the u-spinors: Similarly, we can find the solutions for the v-spinors, which are 4.
The mismatch of subscripts on the two sides is a conventional choice, which we will discuss again in 4. In this frame, we It is easy to check. Since we have multiple solutions, a useful concept is that of the projection operators, which project out the appropriate solutions in some particular situation.
We discuss some such projection operators, which will be quite useful in later parts of the book. I example, if we are interested in states of either positive or negative energy, but not both, we can use the energy projection operators.
Another useful pair are the helicity projection operators. A special direction is the direction of motion of the fermion except in its rest frame, of course. The spin measured along the direction of motion is called helicity, and the helicity projection operators project out states of positive and negative helicities.
We have noted in Eq. Let us therefore define a 3-vector 4. It is also possible to show that for any solution of Dirac equation, the energy projection operators commute with the helicity projection operators,.
These relations are independent of the representation we may choose to use, and this last one shows that we are allowed to choose common eigenstates of A p and II p as the basis of our representation. For example, if the 3-momentum is in the z-direction, one can easily verify that the spinors given in Eqs. For any other direction of the momentum, one can make linear combinations of the two u-spinors of Eq. For the v-spinors, it is the opposite of helicity. As we mentioned after Eq. In general, any operator that squares to the identity can be used to construct a projection operator.
The energy and helicity projection operators A and II are specific examples of this sort of construction. Another obvious operator which squares to the identity is ' '5. These are called the chirality projection operators. As mentioned earlier, helicity projection operators project out definite spin projections along the direction of the 3-momentum of a particle.
They are therefore useless for a definite spin projection of a particle at rest. For that one needs different projection operators. Suppose we want to project the spin states for a certain spatial direction denoted by the unit 3-vector s. For this, first let us define a 4-vector nJl. Notice that nil satisfies the conditions 4. So we can construct the projection operators 4. To understand the significance of these operators, let us consider to be the x-direction, i.
In the rest frame of the. This commutes with the spin operator in the x-direction, El, so the eigenstates of this operator are eigenstates of E' as well. The operators given above project out the two possible values of the spin component. Take the u and the v spinal'S shown in Eq. The problem with a single particle interpretation of the solutions of the Dirac equation is the same as that with a scalar field. It contains negative energy states. So we want to go over to a field theoretic interpretation.
The first step for that is to construct a Lagrangian. The Dirac equation can be derived from a Lagrangian 4. From this, one can derive the Euler-Lagrange equations. We are considering the case where ,p is a complex field see Appendix A. Since components of ,p are linear combinations of the components of ,pI, we can treat ,p and ,p to be independent. Thus, the Euler-Lagrange equation for ,p gives.
Thus we obtain Eq. Quantization of Dirac fields Exercise 4. TToW sign on the den'lJo. The Lagrangian is then also a real function in classical field theory. In quantum physics, fields are operators, so the Lagrangian should be hermitian.
Indeed, the Lagrangians for scalar fields, given in Eqs. But the Dirac Lagrangian of Eq. The mass term is hermitian, but the hermitian conjugate of the term involving derivatives is not equal to itself:. If we really want a hermitian Lagrangian, we can discard the Lagrangian! L' of Eq. The reason is that!
According to the discussion of 2. We will therefore keep using Eq. The Dirac Lagrangian is invariant under the transformations. PIJ is the momentum opera. A theorem then states that, for pa. Using the e: The contra. We now Fourier decompose the Dirac field as we did with the scalar field in the last chapter.
As with the scalar field, we write the Dirac field as an integral over momentum space of the plane wave solutions, with creation and annihilation operators a:: To apply the operator -i'y. V m on 1jJ x given in Eq. Substituting these in Eq. Notice that the cross terms , Le. The other two terms give.
This Hamiltonian can give negative energy eigenvalues, which is a serious problem. And this does not go away by normal-ordering, at least if we assume the definition of normal-ordering employed in Ch. The way to take care of this problem is to assume that the creation and annihilation operators in this case obey anticommutation rather than commutation relations:.
And we define normal ordering for fermionic operators by saying that whenever we encounter products of them, we shall put the annihilation operators to the right and creation operators to the left as if the anticommutators were zero. Using this prescription, then, we obtain the Ilormal-ordered Hamiltonian to be: One can use 0. Moreover, one can define the anticommutation relations of Eq. Show that these quantities must be related b-y. Some consequences of the anticommutation relations and the new definition of normal-ordering are worth exploring at this point.
Just as in the case of complex scalar fields, we see that the Noether charge for the particles created by II is opposite to the charge of those created by ft. The latter are therefore antiparticles of the former ones.
Secondly, we said that all anticommutators other than the ones mentioned in Eq. For example, if we take the anticommutator of fl p with itself, it will vanish.
So will the anticommutator of l1 p with itself. This means that. In other words J we cannot create two particles or two antiparticles with the same spin and same momentum. Anticommutators imply the Pauli exclusion principle. The states containing particles and antiparticles can be constructed by acting on the vacuum by ft and j!. The Hamiltonian of Eq. These imply that both fi k and fi k create positive energy states, a particle in the first case and an antiparticle in the second.
In particular, fi k 10 is a one-particle state, and [,1 k 10 is a oneantiparticle state. To understand the helicities of these one-particle states, we first note that one can use the normalization conditions in Eq.
Here E k is the helicity operator, defined in Eq. For example, the second of these relations gives. If we now use the convention of the helicity eigenstates as in Eq. The helicities of particle states can be calculated in the same way. D Exercise 4. We proceed as we did for the scalar field and write the Dirac equation coupled to a source: Our technique for solving Eq. We take a Fourier transform to write. This is a matrix equation, S p is a matrix. But p2 - m 2 is a number, so we can take it to the denominator on the right hand side.
If we put the Fourier transform obtained in Eq. F X - x' is the propagator for the scalar fields derived in Eq. This gives S , F X -. The matrix elements of SF X - x' can be written in terms of the field operators in the form 4. Like the propagator for the complex scalar field, the propagator in Eq. The S-matrix expansion So far we have dealt with free fields only. A truly free field h"" no experimental consequences, since it would not interact with the detection apparatus at all.
Moreover, if all fields were free fields, they would exist independent of one another and nothing would happen in the world! Of course, the fields in the real world are not free.
They interact with one another, which gives rise to phenomena we can observe. To' describe such phenomena, we therefore need a framework to describe the interactions.
In the Lagrangians of free fields described earlier, we had only terms which are quadratic in field operators. In terms of creation and annihilation operators, this is exactly what is needed for describing free fields. At any space-time point, the free Hamiltonian can annihilate a particle and create the same. Thus, in effect, the I. It is possible to have terms bilinear in two fields which would annihilate a particle of one type and create a particle of another type.
These can be brought around to the standard quadratic form by a redefinition of fields. Alternatively, they can be treated as quadratic interactions of the original fields. To describe any other type of interaction, we need some term in the Lagrangian which has three or more field operators.
Moreover 1 the term has to be Poincare invariant since the Lagrangian density is. Also, the action is a real number, so the Lagrangian has to be. Of course, as in the case of the Dirac field, the Lagrangian can differ from a hermitian Lagrangian by total divergence terms.
The possibilities of interaction depend, of course, on the type of fields in a theory. For example, suppose we have only one real scalar field q, in our theory. Then in addition to the free Lagrangian we can have the following interaction terms:. Here f. J, A etc are some constants which are called coupling constants, Le.
This is because the coupling constants in these terms have negative mass dimensions. Such cou-. If we have a complex scalar field, the interaction terms in Eq. Of course, the Lagrangian must remain hermitian after these replacements. The free Lagrangian of a complex scalar field is invariant under the change of the phase of the field, which was mentioned in Eq.
If for some reason we want to have this symmetry for the interaction terms as well, then the most general form of the interactions would be.
If we have only spinor fields in a theory, our choices are much more restricted. This is because the spinors transform non-trivially under Lorentz transformations 1 and we must make combinations which are Lorentz invariant. A product of an odd number of spinor fields cannot be Lorentz invariant.
Thus the first nOll-trivial combination will. We can also have more than four if we so wish. There can be many types of 4-fermion interactions. For example, we can write an interaction of the form. One combination which played a crucial role in the development of weak interaction theory is.
For example, suppose we want to describe nuclear beta decay in which a neutron decays into a proton, an electron and an electron-antineutrino. In this ease, the last equation turns out to be successful in explaining experimental data provided we replace the four.
The factor of ,J2 in the coupling constant is purely conventional. Defined this way, Gp is called the iJ-decay constant. Notice also that in Eq. This is because the term written down in detail is not hermitian by itself.
Since the Lagrangian has to be hermitian, this means that the hermitian conjugate term must also be present in the Lagrangian. Let us now consider a field theory in which there is one real scalar field and a spinor field ,p. In this case, apart from the interactions. For example, we can have. In the standard model of particle interactions, this kind of interaction arises between the fermions and a scalar boson field representing the Higgs particle.
Similarly, one can have 5. Originally, a term like this was proposed to describe the interaction between protons and neutrons which are together grouped as nucleons since they occur in atomic nuclei with spinless particles called pions,. Exercise 5. In writing all these interactions, we have ignored the free terms in the Lagrangian since they are already known to us.
It is the nature of the interaction that defines a specific theory, and so theories are named after their interaction terms. The type of interaction given in Eq. In the modern literature, interactions of this form and also of the form in Eq. Any theory with a scalar and a fermion field with such interactions is called Yukawa theory. The interaction with four fermionic field operators was first discussed by Fermi and are therefore called Fermi interactions, or sometimes 4-fermion interactions.
Discussion of these and other interactions requires a gClleral framework to deal with interacting fields, which we now proceed to construct. If for a given system we could find all the quantum states we would be able to exactly predict the behavior of the system from its initial conditions.
In other words, given an initial state Ii consisting of. Unfortunately, apart from a few and almost always unrealistic exceptions, it is not possible to find the eigenstates of a Hamiltonian if the fields in it interact with each other. Of course we could have done the separation in the Hamiltonian density as well. Here H o includes all the bits of H whose eigenstates are exactly known to us.
We shall restrict ourselves where H o is the free Hamiltonian of the fields concerned, which can be written in terms of the creation and annihilation operators as described earlier.
The remaining piece, HI, contains all the interactions. We want to find some way of describing the interactions in terms of the free fields. We shall have to resort to perturbation expansions in order to achieve this. There can be situations where H o includes the Hamiltonian for bound states, describing an atom for example, and HI describes interactions of particles with the atom.
We shall not consider these cases, but the formalism we shall develop can be applied equally well to them. On the other hand in the absence of interactions, the evolution of states is governed by the free Hamiltonian H o, which is constructed out of free field operators.
The states then obey the equation. If the states on both sides are taken to be normalized states, the operators Uo t and U t are unitary for any t. The initial state Ii. We can write Eq. Let us now consider a system evolving under the full Hamiltonian. SchrOdingerls equation for this system is. The first term on the left cancels the first term on the right, and multiplying both sides by Ud t , we get. The Hilbert space on which both sides of Eq.
These are then the time-independent free states of definite quantum numbers. Here we have written. In other words, RI t is the interaction Hamiltonian Hll now written in terms of free fields at time t. Sometimes this is referred to as the 'interaction Hamiltonian in the interaction picture'. We could think of this as saying that all fields are free fields in the remote past, and the interactions.
This makes sense in scattering experiments, where we have particles moving in as free particles, interacting briefly, and moving away as free particles again. Our assumption boils down to U t -. We can obtain a solution iteratively if we replace the U tIl on the right hand side by Eq.
This gives. Note that. Now, by interchanging the dummy variables tl and t2 in the second integral and adding up the two integrals, this reduces to 5. Similarly, the higher terms can also be rewritten in terms of timeordered products, and the perturbative solution for U I can be writ! The expre"iou for U I. As a result, we arc usually int. Here, as in qlla,ntullI Illc 'hanicti, 'small' implies t.
Jm]Jt tic series, one ill which c: One could therefore question the validity of the formalism derived above, fl. This is not a t. It is trllly f0.
We have already shown in Eq. Thus the S-matrix is unitary:. Why is the S-matrix useful? Particle interactions usually occur for small intervals of time. Long before thb time, one can consider the particles to be essentially free. Similarly, long after the interaction period, the particles are again essentially free. A typical problem of interaction can then be reduced to finding out whether a given state of free particles, through interactions, can evolve to another given state of free particles long after all interactions have cea.
Let us denote this initial state by Ii. Now let us slowly turn on the interaction, let tlw state evolve under the full Hamiltonian H, and again slowly turn off the interaction. After a long time the system must again be free and we call describe it by a superposition of states of the form Uo t If! Using Eq. The 'in' and 'out' states are separately taken to be orthonormal and complete.
The S-matrix is defined as the infinite matrix with elements 5. Then the elements of the S-matrix are. There are certain assumptions involved in taking these limits, but for all purposes we can ignore such subtleties and work with the S-operator. The normal ordering procedure involved putting all the annihilation operators to the right of all the creation operators so that it annihilates the vacuum. But the time ordering raises complications because in it all operators at earlier times must be further to the right.
So creation operators at earlier times would be to the right of annihilation operators at later times, contrary to what we need for normal ordering.
The advantage of normal ordered products is that their expectation values vanish in the vacuum. So we would. For this, we need a relation between the two. This relation is provided by Wick's theorem. Consider first the simple example of the time-ordered product of two foctors of a scalar field.
Remember the definition given in Eq. Our task reduces to writing the simple products on the right hand side in terms of normal ordered products. To this end, let us write. They are thus shorthands for the two terms appearing in Eq. Because of the definition of the vocuum state in Eq. On the other hand, if we write the normal ordered operator: In other words,. S 47 The commutator on the right side can be put into a more convenient form. Because of Eq. We can also write it as. However, since the commutator appearing on the right side is a number, its vacuum expectation value is the number itself.
Thus we can write. On the right side here, we have the vacuum expectation value of the time-ordered product of two field operators. Since such quantities will appear often in the ensuing discussion, it is useful to develop a compact notation for them. Such quantities are usually called Wick contractions and are denoted by the following symbol:. More generally when the two fields appearing in the prod net are not necessarily the same, we will write.
Since the contraction is a vacuum expectation value, it will vanish unless one operator to the right creates a particle which the other. So if they correspond to different fields, this expectation value vanishes.
If they correspond to the same field or a field and its adjoint we get i times the respective Feynman propagators,. Wick's theorem can be generalized to any number of field operators by multiplying on the right by a field and considering the interchanges required to change from a time ordered product to a normal ordered product. The theorem is then proven by induction. The result is. A few words about the notation used in writing up this result. All possible pairs are contracted, first one pair, then two pairs, and so on until we run out of pairs to contract.
The word 'perm I occurring here means permutations on all fields present. The formula is long. However, since the contractions are not field operators but rat.
Finally, in the S-matrix expansion, we had time-ordered products of normal-ordered objects: We need to avoid commutators or anticommutators at the same point in space-time because they are infinite.
Then within each normal ordered group we have automatic time ordering so there is no contraction within a group. Wick's theorem looks formidable, but actually its implementation is not very difficult. In calculating S-matrix elements for specific initial and final states, we need to take the matrix elements of the time-ordered product given above between those states.
Thus in the Wick expansion, we need only those terms where, apart from the contractions, we have just the right number of field operators to annihilate the initial state and create the final one. These comments will probably be clearer as we look at specific examples. The theorem is then 'P1'o13ed b'Y induction, starting from Eq. From Wick expansion to Feynman diagrams In this chapter, we show how to use the Wick expansion to calculate S-matrix elements involving scalars and spinors.
As a concrete example, we take a theory with a fermion field and a scalar field, which interact via the Yukawa interaction:. We will consider specific initial and final states, and in each case see which terms in the S-matrix expansion can give non-zero contribution to the relevant S-matrix element. An alternative approach could involve considering a specific order in the S-matrix expansion and investigate which processes it can contribute to.
This approach will be taken later in the book for Quantum Electrodynamics. The quanta of the fermionic field 1JJ will be called electrons. We give the names only to avoid cumbersome sentences when we explain what is going on.
Let us denote the mass of the particle B by M and the mass of the electron by m. Our objective is to calculate the S-matrix elements for this process. The Hamiltonian derived from the Lagrangian will also have the usual free terms, minus the interaction Lagrangian. This is obviously true for all theories without derivative interactions.
It also happens to be true if derivative interactions are present. For the fermionscalar interaction,. Let us now look at the term linear in the interaction Hamiltonian in the S-matrix. In the last step, we have omitted the time-ordering sign, since in this case, there is only one space-time point involved and so there is just one time. Consequently, Wick's theorem is not needed here - we already have the normal ordered product. There are many exercise problems to help the students, instructors and beginning researchers in the field.
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Hardcover Verified download. This book by two Indian professors from Culcutta is concise, and focused on getting you from quantum mechanics to QFT in the fastest possible way. In that sense it is well suited for those folks who have taken a quantum mechanics class and are impatient to understand QFT.
The binding is not that durable, after few weeks of use, some of the pages have already shifted. Unforunately, the material and specially the derivations are not well explained. For example, I found it frustrating to go through the derivation of Noether's theorem in chapter 2. For comparison, it is roughly at the same difficulty level as Ryder and Maggiore. Ryder can be very verbose. Explanations in Ryder are sometimes excellent but sometimes just bad.
Maggiore on the other hand is concise, consistent and to the point. Of all the truly introductory QFT books, in my opinion, Maggiore is the hands down winner. I have about ten books on Quantum Field Theory. It is a very hard subject to learn. Each one has some good points, but this one is uniformly good. It is an introduction, so eventually, you will need other texts. I had to specially order this book but it was well worth it. To start the authors presuppose only the standard undergraduate mathematical background.
Readers adept in multi-variable calculus and linear algebra with applications to Special Relativity and introductory quantum mechanics will have no difficulty with the book.
They first introduce classical field theory and develop both the Langrangian and Hamiltonian concepts culminating in the principle of least action.
Variational calculus techniques are employed to develop the standard euler langrange equations. Noether's theorem is introduced which roughly speaking, states that for each symmetry of the Langrangian one has a conserved quantity. For example space translation invariance leads to conservation of linear momentum and time symmetry leads to conservation of energy.
Two types of symmetries are distinguished, namely internal and external. A first course in general relativity would provide all the special relativity required as well as giving some geometric perspective of gauge theories. In principle a student can fit all of this in by the end of their sophomore year. For over a decade this has been the standard in QFT pedagogy.
It has everything a student could want presented coherently. The style is geared towards calculations, which makes it a handy reference. What I like about it: The book works remarkably well for a meticulous line-by-line study. More advanced topics are covered very well, including a very good treatment of the renormalization group Peskin was a student of Ken Wilson.
Conceptual chapters that begin each part are especially well written. Conclusion: Every QFT student needs this book… but absolute beginners may not find this the most user-friendly text and would benefit from additional references. Zee, ASTI lectures. Zee, Quantum Field Theory in a Nutshell. Peppered with anecdotes and flavored with a playful writing style, QFT in a Nutshell is the entree of the meal. What I like about it: The writing style and presentation of topics is nearly Feynman-esque in its clarity and pedagogy.
It is written at just the right level for students who want to build up a solid conceptual understanding. What confused me the first time I read it: In any QFT course, students must learn QFT twice: once using the canonical formalism and once using the path integral formalism.
Peskin teaches them in that order. Unfortunatly Zee teaches in the opposite order. This makes it difficult to synchronize a reading of Peskin and Zee until one has done at least few chapters of one or the other. The canonical formalism follows from quantum mechanics more readily, while the path integral formalism is more physically transparent.
To bridge the conceptual gap between Griffiths and Peskin, Zee hits the spot quite well. If one has plenty of time and is in no rush to calculate diagrams, consider reading all of Part I of Zee, especially the first parts and section I.
Griffiths, Introduction to Elementary Particles. This reading is very accessible and can be completed rather quickly. Chapter 2 provides a glimpse of the Standard Model and Feynman diagrams.
Chapters 4 and 5 on symmetries and bound states can be skipped. Chapter 6 begins to reveal the machinery of QFT, explaining how to calculate simple cross sections and Feynman rules. Chapter 7 explains how this all works for fermions, including an introduction to the Dirac equation that is excellent preparation for the analogous chapter in Peskin.