Energy and States of Particles

In possession of the concept of energy within the lattice formalism, we will now discuss in considerable more detail the concept of state. This discussion will lead us to the construction within this formalism of states with a given number of particles and, therefore, to the concept of particle itself. This is a central concept within the structure of the theory, which is directly connected with the fundamental issues of the observation of physical phenomena and of the process of measurement.

Up to now we have been defining and developing a formalism that allows us to define and calculate, at least in principle, any observables within any given model of quantum field theory. We have been doing this through the use of a statistical model in which we define a certain statistical distribution of probabilities that applies to each and every one of the possible configurations of the fields. This distribution is the Boltzmann distribution, which can be expressed in terms of the action functional $S$ of a given model as

$$\vert\rangle\sim\frac{\displaystyle [{\rm d}\varphi]\;e^{-S[\varphi]}}{\displaystyle \int[{\rm d}\varphi]\;e^{-S[\varphi]}},$$

or, if we wish to use the canonical formalism, as

$$\vert\rangle\sim\frac{\displaystyle [{\rm d}\varphi][{\rm d}... ...m_{s}[\bar\pi \Delta_{0}\varphi-{\cal H}(\varphi,\bar\pi )]}}.$$

The symbolism of Dirac “bras” and “kets” that we use here expresses the fact that our interpretation of the structure that we are building is that this statistical distribution is a representation of the vacuum state of the theory, for this particular model.

It is a difficult task to define a-priori what the vacuum might be. The classical idea that it is a situation in which there is total absence of any physical content is not very useful in the context of the quantum theory, due to the concept of uncertainty that is inherent to such a theory. We will simply say that this state defines completely the physical situation in the region of space-time where it is realized, and accept its introduction as part of the definition of the quantum theory. We will also see that it is a state that contains no particles, that is, no observable amount of energy. This does not mean that there is nothing in it, because there is the field, which fluctuates permanently in a rather violent way. One might say that the vacuum is the state that contains nothing but the minimum mount of uncertainty which is inherent to the quantum theory.

This idea which we introduced above, that physical states are connected with certain statistical distributions, immediately suggest the generalization of its application to other stated besides the vacuum. Strictly speaking this is not necessary for the measurement of observables, since we can measure any observables using only the vacuum state. This is a remarkable characteristic of this structure of ours, it is enough to define a single state in order for us to be able to define and calculate all the relevant observables of the theory, that is, all the correlation functions and any other observables, related to other functionals of the fields. However, the introduction of the direct representation of other states enriches our structure and permits a better understanding of its functioning.

In this section we are using the word “state” with a very general meaning, as a representation of the physical situation in a given region of space-time. We are going to make here no attempt to establish a definite formal relation with the concept of states as vectors in a Hilbert space. In fact, we are not going to talk at all about Hilbert spaces or the operators that exist in these spaces. We are going to talk only about physical states and observables. Later on we will see to what extent it is possible to establish a relation between our structure and the Hilbert spaces of quantum mechanics.

Very well, based on the experience we have with the traditional formalism it is not difficult to guess at the form that a one-particle state should have. Pushing ahead the connection between states and statistical distributions, we introduce the state of one particle with momentum $\vec{k}$ through the definition of a new statistical distribution of configurations,

$$\vert 1,\vec{k}\rangle\sim\frac{\displaystyle [{\rm d}\varph... ...]\;\vert\widetilde\varphi _{\vec{k}}\vert^{2}e^{-S[\varphi]}},$$

or, in terms of the canonical formalism,

$$\vert 1,\vec{k}\rangle\sim\frac{\displaystyle [{\rm d}\varph... ...m_{s}[\bar\pi \Delta_{0}\varphi-{\cal H}(\varphi,\bar\pi )]}},$$

expressions where there appears the Fourier component of the field $\varphi$ associated to the momentum-space mode $\vec{k}$. Observe that any expectation value of an observable on this state can be reduced to the ratio of two expectation values on the vacuum, by the simple division of both numerator and denominator by the normalization factor of the vacuum distribution,

$$\langle{\cal O}\rangle_{1,\vec{k}}=\frac{\displaystyle \left... ...\langle\vert\widetilde\varphi _{\vec{k}}\vert^{2}\rangle_{0}},$$

where the index $0$ on the expectation values indicates that they are taken on the vacuum state. Hence we see that in fact the vacuum is sufficient for the calculation of any observables, a fact which is of great important, for example, to permit the computational calculation of the expectation values of observables on other states by reduction to expectation values on the vacuum state.

Figure: Qualitative diagram of the probability distribution $\exp(-x^{2})$ of the Fourier components of the field in the vacuum state, showing the point of maximum at $0$.

Before we begin to examine some properties of this new distribution let us emphasize here that, while we are proposing a relation between states of the quantum theory of fields and statistical distributions of the fields, we are absolutely not stating that any such statistical distribution is related to a physical state of the theory. There are many distributions that are clearly not related to physical states, such as, for example, any “delta-functional” distribution, that attributes the probability $1$ to a certain configuration and the probability $0$ to all others, because this would translate into a physical situation in which the fundamental field does not fluctuate at all, which is a classical, not a quantum situation. We will postpone to a future opportunity a more detailed discussion of the conditions that the distributions must satisfy in order to be associated to states, and will limit ourselves here only to the comment that such conditions are related to the principle of uncertainty and to the issues of observation and measurement. Our difficulties with the “ab-initio” definition of the vacuum will have to be resolved in the context of this future discussion, possibly through the criterion that the vacuum is the lowest-energy state that satisfies such conditions.

Figure: Qualitative diagram of the probability distribution $x^{2}\exp(-x^{2})$ of the Fourier component of the field that is singled out in the one-particle state with a given momentum vector $\vec{k}$, showing the point of maximum at $1$.

For the time being, we will limit ourselves to the examination of distributions containing the Boltzmann factor $\exp(-S)$ and powers of the Fourier components of the fields. Note that the effect of the introduction of the factor $\vert\widetilde\varphi _{\vec{k}}\vert^{2}$ in the distribution is intuitively clear. While the distribution given by the exponential $\exp(-S)$, where $S$ is quadratic on all the Fourier components, concentrates the probabilities around the value $0$, where it has its maximum value, as one can see in the graph of figure 5.2.1, the introduction of the factor $\vert\widetilde\varphi _{\vec{k}}\vert^{2}$ causes the displacement of this point of maximum to a finite and non-vanishing value, around which the probabilities become concentrated, as shown in the graph of figure 5.2.2. We will see that in the case of the state of $n$ particles this maximum will be displaced to a value proportional to $\sqrt{n}$. Since this happens only for the part of the distribution related to the mode $\vec{k}$, through the introduction of the factor $\vert\widetilde\varphi _{\vec{k}}\vert^{2}$ we are favoring the configurations that have a larger component of plane wave with momentum $\vec{k}$.

In order to calculate the energy of this new state, we will use the canonical definition and the form of the dimensionless Hamiltonian defined in section 5.1,

$$\mathbf{H}=-\frac{\imath}{2}\sum_{\mathbf{x}}\left[\bar\pi ^... ...+\sum_{i}(\Delta_{i}\varphi)^{2}+\alpha_{0}\varphi^{2}\right].$$

With these ingredients we obtain (problem 5.2.1) for the expectation value of the Hamiltonian in the state of one particle with momentum $\vec{k}$,

$$\langle\mathbf{H}\rangle_{1,\vec{k}} =-\frac{\imath}{2}N_{L}... ...]\;\vert\widetilde\varphi _{\vec{k}}\vert^{2}e^{-S[\varphi]}},$$

where $\mathbf{H}'$ is, as before, the expression of $\mathbf{H}$ with $\bar\pi$ substituted by $\imath\Delta_{0}\varphi$, showing once more that the difference between the canonical definition and the initial definition is just a constant that diverges in the continuum limit. Repeating procedures used before in section 5.1 we may write explicitly for the energy

$$\begin{eqnarray*} \imath E_{1,\vec{k}}T & = & \frac{N_{T}N_{L}^{d-1}}{2} \\ & & ... ...widetilde\varphi _{\vec{k}}\vert^{2}e^{-S[\widetilde\varphi ]}}, \end{eqnarray*}$$

where we recall that we have for $S[\widetilde\varphi ]$ written in momentum space

$$S[\widetilde\varphi ]=\frac{N_{T}N_{L}^{d-1}}{2} \sum_{\vec{... ...k}}^{2}+\alpha_{0})\vert\widetilde\varphi _{\vec{k}}\vert^{2},$$

with

$$\rho_{\vec{k}}^{2}=4\sin^{2}\!\left(\frac{k_{0}\pi}{N_{T}}\right) +\sum_{i}4\sin^{2}\!\left(\frac{k_{i}\pi}{N_{L}}\right).$$

For the terms of the sum over the momenta $\vec{q}$ such that $\vec{q}\neq\pm\vec{k}$ the calculation proceeds as in the case of the calculation of the energy of the vacuum, but for $\vec{q}=\pm\vec{k}$ there are differences, because in this case we have the expectation value of a larger power of the Fourier component $\widetilde\varphi _{\vec{k}}$ of the fields. Taking into account the symmetries by exchange of the sign of $\vec{q}$ in the sum, we may write

$$\begin{eqnarray*} \imath E_{1,\vec{k}}T & = & \frac{N_{T}N_{L}^{d-1}}{2} +\frac{... ... {\langle\vert\widetilde\varphi _{\vec{k}}\vert^{2}\rangle_{0}}, \end{eqnarray*}$$

where we already used in the second term the result

$$\langle\vert\widetilde\varphi _{\vec{k}}\vert^{2}\rangle_{0}... ...1}{N_{T}N_{L}^{d-1}}\;\frac{1}{\rho_{\vec{k}}^{2}+\alpha_{0}}.$$

For the other expectation value, which appears in the third term, we have

$$\langle\vert\widetilde\varphi _{\vec{k}}\vert^{4}\rangle_{0}... ...}} \;\frac{1}{\left(\rho_{\vec{k}}^{2}+\alpha_{0}\right)^{2}},$$

where we used the factorization relations given in section 3.4, for the case $\vec{k}\neq\vec{0}$, since the case $\vec{k}=\vec{0}$ would correspond to particles without any energy and without any $(d-1)$-dimensional momentum, being therefore of no interest. We may use these results in the third term and reorganize the terms in order to complete the sum of the second term in such a way that it runs over all possible values of $\vec{q}$, obtaining, after some manipulation,

$$\imath E_{1,\vec{k}}T =\sum_{\vec{q}}\frac{\sum_{i}\rho_{i}^... ...ho_{0}^{2}(\vec{k})+\sum_{i}\rho_{i}^{2}(\vec{k})+\alpha_{0}}.$$

One observes here that the first term is precisely the energy of the vacuum $E_{0}$, a quantity that diverges in the continuum limit. We may now define the quantity

$$\Delta E_{1,\vec{k}}=E_{1,\vec{k}}-E_{0},$$

in which we subtracted from the energy its value in the vacuum state, obtaining

  $$\imath\Delta E_{1,\vec{k}}T =\frac{-\rho_{0}^{2}(\vec{k})+\sum_{i... ...+\alpha_{0}} {\rho_{0}^{2}(\vec{k})+\sum_{i}\rho_{i}^{2}(\vec{k})+\alpha_{0}}.$$ (5.2.1)

Observe that this definition makes irrelevant the difference between the canonical definition and the usual definition, since this difference will always cancel out in the expression of $\Delta E$. It is this quantity, the additional energy with respect to the energy of the vacuum that is contained within the state, that we will interpret as the physical energy to be associated to the state. This is equivalent to saying that the observable associated to the physical energy is a modified Hamiltonian,

$$\Delta\mathbf{H}=\mathbf{H}-\langle\mathbf{H}\rangle_{0},$$

so that the dimensionless physical energy is given in terms of the expectation values of this observable,

$$\Delta{\cal E}=\langle\Delta\mathbf{H}\rangle,$$

on any state, while the dimensionfull energy is related to this dimensionless quantity by $\Delta E=\Delta{\cal E}/a=N_{T}\Delta{\cal E}/T$.

As we will see later on, we may extend the definition of our states to arbitrary numbers of particles but, before we do that, let us discuss the physical meaning of the expression we obtained for the energy in the continuum limit. Our definition of the one-particle state is in fact the definition of a collection of states, one for each $d$-dimensional mode $\vec{k}$ existing on the lattice, each one of them having its energy given in terms of $\vec{k}$ by the expression in equation (5.2.1). If we write the version of this result in Minkowski space, thus de-Euclideanizing the result, we obtain

$$\Delta E_{1,\vec{k}}=\frac{-1}{T}\; \frac{\rho_{0}^{2}(\vec{... ...ho_{0}^{2}(\vec{k})+\sum_{i}\rho_{i}^{2}(\vec{k})+\alpha_{0}}.$$

There are, in fact, two limits to discuss here, the continuum limit in which we make $N\rightarrow \infty$, and the $T\rightarrow \infty$ limit in which we make the box infinite in the temporal direction. We will discuss the first limit in the symmetrical case, making $N_{T}=N_{L}=N\rightarrow\infty$ while we keep $T$ finite, leaving for later on the discussion of other ways to take this limit. In this case we may multiply both the denominator and the numerator by $N^{2}/L^{2}$, take the limit and write the result as

$$\Delta E_{1,\vec{k}} =\frac{-1}{T}\;\frac{p_{0}^{2}(\vec{k})... ...^{2}} {-p_{0}^{2}(\vec{k})+\mathbf{p}^{2}(\vec{k})+m_{0}^{2}},$$

where $m_{0}^{2}=\alpha_{0}N^{2}/L^{2}$ and the $(d-1)$-dimensional momentum $\mathbf{p}$ is defined by

$$\mathbf{p}_{\vec{k}}^{2} =\lim_{N\rightarrow\infty}\frac{N^{... ...\pi}{N}\right) =\sum_{i}\left(\frac{2\pi k_{i}}{L}\right)^{2}.$$

Let us consider now the limit $T\rightarrow \infty$, with $L$ either kept fixed or not. Since $T$ appears in the denominator, our expression for the energy of a particle goes to zero, unless the momentum-dependent expression in the denominator vanishes in the limit. This takes us to the on-shell condition, that selects a subset of all possible $d$-dimensional modes. In order to see this we may rewrite the expression as

$$\Delta E_{1,\vec{k}} =\frac{-1}{T}\;\frac{p_{0}^{2}(\vec{k})... ...(\vec{k})+\sqrt{\mathbf{p}_{\vec{k}}^{2}+m_{0}^{2}}\,\right]}.$$

Observe that we can obtain a finite and non-vanishing limit only so long as in the limit one of these two relations holds,

$$p_{0}(\vec{k})=\sqrt{\mathbf{p}_{\vec{k}}^{2}+m_{0}^{2}}\mbo... ...~~} p_{0}(\vec{k})=-\sqrt{\mathbf{p}_{\vec{k}}^{2}+m_{0}^{2}}.$$

We thus obtain the on-shell condition that relates the energy, the momentum and the rest mass of a relativistic particle. We see also that we may have some limits in which the energy is positive as well as other limits in which it is negative, as mentioned in section 5.1. Besides the two possibilities presented by the two factors in the denominator, in each case it is possible to take the limit in which $p_{0}$ approaches $\pm\sqrt{\mathbf{p}^{2}+m_{0}^{2}}$ either by smaller values or by larger values, thus changing the sign of the energy. The issue of the positiveness of the energy will remain open here because it cannot be solved in a theory of electrically neutral particles with spin zero as is the case for the real scalar fields we use here as an example. The resolution of this problem will have to wait until we are able to introduce into the structure of the theory other essential elements.

Observe that if our system is inside a box in which both $T$ and $L$ are finite then it may not be possible to satisfy an on-shell condition such as this one for arbitrary values of the mass $m_{0}$, because in this case both the values of $p_{0}$ and the values of $\mathbf{p}$ are quantized at discrete values, and there is no continuous variable except the mass that we may vary so that the equality can be satisfied. This problem disappears when we make $T$ go to infinity, as we must, since in this case $p_{0}$ becomes a variable that can be varied continuously, and therefore it is always possible to satisfy the on-shell condition by varying $p_{0}$. If in addition to the limit $T\rightarrow \infty$ we also take the limit $L\rightarrow\infty$ then both $p_{0}$ and $\mathbf{p}$ become continuous variables and we obtain the usual on-shell condition for particles in infinite space-time.

If we keep $L$ finite then the discrete character of $\mathbf{p}$ will be reflected, through the on-shell condition, on a corresponding discretization of the values of $p_{0}$. Thus we see here a simple example of the mechanism that leads to the appearance of energy quantization for bound states, which are confined to a finite region of the $(d-1)$-dimensional space. Note that making $T\rightarrow \infty$ while $L$ is kept fixed is equivalent to taking the non-relativistic limit, since with $T\gg L$ only phenomena involving very small velocities will have world-lines that fit into the $d$-dimensional box. We therefore see here a very important fact, that the interpretation of relativistic particles as excitations of the modes of the $d$-dimensional cavity is reduced, in the non-relativistic limit, by means of the on-shell condition, to the association of physical particles to the energies and modes of the corresponding $(d-1)$-dimensional spacial cavity.

Adopting arbitrarily the first of the two possibilities above, we may impose that the $T\rightarrow \infty$ limit be taken in such a way that we have in this limit

$$T\left[-p_{0}(\vec{k})+\sqrt{\mathbf{p}_{\vec{k}}^{2}+m_{0}^{2}}\,\right]=A,$$

for some finite, dimensionless and constant number $A$, so that

$$p_{0}(\vec{k})=-\frac{A}{T}+\sqrt{\mathbf{p}_{\vec{k}}^{2}+m_{0}^{2}}.$$

We see here that, for finite $T$, the on-shell condition is modified, that is, that the energy of each mode is modified by a term proportional to $1/T$, exactly as we verified for the energy of the vacuum in the case of quantum mechanics. This is, therefore, an infrared effect due to the finite size of the temporal box, exactly as before. Note that this comparison to the quantum-mechanical case already seems to indicate that the natural value for $A$ is $-1$. We may now substitute this relation for $p_{0}(k)$ in the expression of the energy, obtaining in the $T\rightarrow \infty$ limit

$$\Delta E_{1,\vec{k}}=\frac{-1}{A}\sqrt{\mathbf{p}_{\vec{k}}^{2}+m_{0}^{2}}.$$

We see therefore that, in order for the expectation value of the energy to coincide numerically with the temporal component of the vector $\vec{k}$, we should impose that the $T\rightarrow \infty$ limit be such that $A=-1$. Note that this arbitrariness in the value of $A$ is equivalent to the arbitrariness in the choice of units for the energy.

It is interesting to discuss here the case $d=1$ and thus verify that we obtain the correct results for the harmonic oscillator in quantum mechanics. In this case the on-shell condition within a finite temporal box reduces, already making $A=-1$, to

$$p_{0}=\frac{1}{T}+m_{0},$$

that is, except for the infrared effects due to the finite size of the temporal box, the energy parameter $p_{0}$ reduces to the mass parameter $m_{0}$. Since in this case we do not have the components $p_{i}$, this relation determines completely $p_{0}$ and, therefore, $k_{0}$. If we recall that we have $\alpha_{0}=m_{0}^{2}a^{2}$ and that this same parameter $\alpha_{0}$ relates to the angular frequency $\omega$ of the harmonic oscillator by $\alpha_{0}=\omega^{2}a^{2}$, we see that we have $m_{0}=\omega$, so that we may write

$$p_{0}=\omega+\frac{1}{T}=\omega\left(1+\frac{1}{\omega T}\right).$$

We may now substitute this value for $p_{0}$ in the expression of the expectation value of the Hamiltonian, obtaining, after some manipulation and keeping only the first-order corrections in $1/T$, in the limit of very large $T$,

$$\Delta E_{1}=\omega\left(1+\frac{1}{2\omega T}\right),$$

showing that this quantity also suffers infrared deviations, is a way similar to $p_{0}$. With a slightly different choice for $A$, making $A=-1+1/(2\omega T)$ rather than $A=-1$, we can make $p_{0}$ and $\Delta E_{1}$ approach their limits in exactly the same way. In any case, in the $T\rightarrow \infty$ limit we have the result

$$\Delta E_{1}=\omega,$$

which is the correct result for the difference between the energies of the first excited state and of the fundamental state of a one-dimensional harmonic oscillator in quantum mechanics.

We may now extend our definition of particle states to arbitrary numbers of identical particles. The state of $n$ particles with momentum $\vec{k}$ can be defined by means of the distribution

$$\vert n,\vec{k}\rangle\sim\frac{\displaystyle [{\rm d}\varph... ...\;\vert\widetilde\varphi _{\vec{k}}\vert^{2n}e^{-S[\varphi]}},$$

or, in terms of the canonical formalism,

$$\vert n,\vec{k}\rangle\sim\frac{\displaystyle [{\rm d}\varph... ...m_{s}[\bar\pi \Delta_{0}\varphi-{\cal H}(\varphi,\bar\pi )]}}.$$

One may now calculate the energy (problem 5.2.2), obtaining, as physically expected, the result

$$\imath\Delta E_{n,\vec{k}}T=n\;\frac {-\rho_{0}^{2}(\vec{k})... ...ho_{0}^{2}(\vec{k})+\sum_{i}\rho_{i}^{2}(\vec{k})+\alpha_{0}}.$$

We see therefore that we obtain in fact a “ladder” of states, whose energies are integer multiples of a finite quantity, even on finite lattices, where both the dimensionless quantities $N_{T}$ and $N_{L}$ and the dimensionfull quantities $T$ and $L$ are finite. This ladder survives the continuum limit within an infinite temporal box so long as the on-shell condition is satisfied in the limit. For modes of the lattice that do not satisfy the on-shell condition the ladder collapses in the continuum limit and its steps become of vanishing height, so that all the collection of states related to it becomes energetically degenerate with the vacuum, not corresponding therefore to states of physically observable particles.

It is interesting to note here that the existence of this ladder of energies for the particle states in the non-linear $\lambda\varphi^{4}$ model, in $d=4$, can be verified directly without too much difficulty by numerical means [3]. The rules for the construction of the Hamiltonian and of the particle states in the case of that model are exactly the same that we used here, the results being different, of course, but only due to the different form of the action. The measurement of the energies of the particle states on finite lattices can be made with great precision, leading to a very precise verification of the proportionality of the energy with the number $n$ of particles. However, the determination of the energy of a particle in the continuum limit and the verification of the on-shell condition are much more difficult from the computational point of view and so far have been done only in a very rough and qualitative way. Whether or not the existence of such ladders of states is related to the phenomenon of the triviality of that model is currently unknown.

We will end this section showing that there exists in our structure an observable that gives, as its expectation values, the number of particles of a given state. This turns out to be the action of the model itself, which functions as a “number of particles” observable, so long as we subtract from it its expectation value on the vacuum, in analogy with what we did for the energy. Recalling once more that the form of the action of our free model in momentum space is

$$S[\widetilde\varphi ]=\frac{N_{T}N_{L}^{d-1}}{2} \sum_{\vec{... ...k}}^{2}+\alpha_{0})\vert\widetilde\varphi _{\vec{k}}\vert^{2},$$

it is easy to calculate directly its expectation value on the vacuum, which has already been done as a problem proposed in a previous section, with the result

$$\langle S\rangle_{0}=\frac{N_{T}N_{L}^{d-1}}{2}.$$

One might interpret this result as one half the number of degrees of freedom of the $d$-dimensional lattice, but this is of no direct physical importance. What is most interesting to us is the calculation of the expectation value of $S$ on the state of one particle with momentum $\vec{k}$, which can easily be done (problem 5.2.3), resulting in

$$\langle S\rangle_{1,\vec{k}}=\frac{N_{T}N_{L}^{d-1}}{2}+1,$$

and, in general, on a state of $n$ particles with momentum $\vec{k}$

$$\langle S\rangle_{n,\vec{k}}=\frac{N_{T}N_{L}^{d-1}}{2}+n,$$

Hence, the observable

$${\cal N}=S-\langle S\rangle_{0}$$

gives us the number of particles of a given state. This can be extended to states for arbitrary numbers of particles with various different momenta, in which case it gives us the total number of particles (problem 5.2.4). Note that this observable is not sensitive to whether or not the particles correspond to modes that satisfy the on-shell condition.

It is interesting to note here that the definition of this observable is quite general and does not depend on any particularity of our simple model here. In fact, it is possible to verify numerically that the observable ${\cal N}$ gives us the number of particles even in non-linear models such as, for example, the $\lambda\varphi^{4}$ model. Not only one verifies that the expectation value of this observable is always proportional to the number $n$ of particles, whatever the values of the parameters of the model may be, but one also verifies that the value of the increment $\Delta\langle{\cal N}\rangle$ between the states of $n$ and $n+1$ particles approaches the value $1$ in the immediacy of the critical region in the space of parameters of the model, indicating that this increment tends to $1$ in the continuum limit. The existing numerical results, still of a somewhat limited quality due to the limitations of the available resources, can be found in [3].

In a linear model such as our standard example here it is possible to define, additionally, observables that function like projection operators, returning the number of particles with a given momentum $\vec{k}$ that exist on the state. The definition of these observables is simple,

$${\cal N}_{\vec{k}}=\frac{1}{2}\left[N_{T}N_{L}^{d-1} \left(\... ...{0}\right)\vert\widetilde\varphi _{\vec{k}}\vert^{2}-1\right].$$

It is easy to verify (problem 5.2.6) that we have for this observable

$$\langle{\cal N}_{\vec{k}}\rangle_{n,\vec{k}}=n,$$

while for $\vec{q}\neq\pm\vec{k}$

$$\langle{\cal N}_{\vec{k}}\rangle_{n,\vec{q}}=0,$$

showing that the observable is, in fact, a projector for particles with momentum $\vec{k}$. Using these observables one can, for example, separate real particles, corresponding to modes satisfying the on-shell condition, from virtual particles corresponding to other modes. In non-linear theories it is not clear whether or not it is possible to define observables like this one in a general way.

Observe that we are not able to distinguish states of particles with momentum $\vec{k}$ from states of particles with momentum $-\vec{k}$, both with respect to the number of particles and with respect to the energy. This is due to the real nature of the scalar field of our simple model, which corresponds to particles without electrical charge. Both with respect to the positivity of the energy of the physical states and with respect to the complete definition of the observables that give us the number of particles, it is clear that, in order to go ahead with the physical interpretation of the theory, it would be necessary to introduce into it complex fields corresponding to charged particles, as well as the gauge fields of electrodynamics.

Problems

1. Calculate the expectation value of the Hamiltonian $\mathbf{H}$ on the state of one particle with momentum $\vec{k}$. During the calculation consider carefully the cases in which $\widetilde\varphi (\vec{k})$ is real and those in which $\widetilde\varphi (\vec{k})$ has a non-vanishing imaginary part.

2. Calculate the expectation value of the Hamiltonian $\Delta\mathbf{H}$ on the state of $n$ particles with momentum $\vec{k}$. During the calculation consider carefully the cases in which $\widetilde\varphi (\vec{k})$ is real and those in which $\widetilde\varphi (\vec{k})$ has a non-vanishing imaginary part.

3. Calculate the expectation value of the action $S$ on the state of $n$ particles with momentum $\vec{k}$. During the calculation consider carefully the cases in which $\widetilde\varphi (\vec{k})$ is real and those in which $\widetilde\varphi (\vec{k})$ has a non-vanishing imaginary part.

4. Calculate the expectation value of the observable ${\cal N}$ on a state having $n_{1}$ particles with momentum $\vec{k}_{1}$ and $n_{2}$ particles with momentum $\vec{k}_{2}$, which is obtained by multiplying the Boltzmann factor by the appropriate factors involving the Fourier components of the fields relative to these two momenta,

   
   

   $$\vert n_{1},\vec{k}_{1};n_{2},\vec{k}_{2}\rangle\sim\frac{\d... ...widetilde\varphi _{\vec{k}_{2}}\vert^{2n_{2}}e^{-S[\varphi]}}.$$
   
   

5. Calculate the expectation value of the Hamiltonian $\Delta\mathbf{H}$ on the state having $n_{1}$ particles with momentum $\vec{k}_{1}$ and $n_{2}$ particles with momentum $\vec{k}_{2}$ considered in problem 5.2.4. Show in this way that the energy is additive, that is, that the total energy of the state is the sum of the energies of the particles that it contains.

6. Calculate the expectation value of the observable ${\cal N}_{\vec{k}}$ on a state having $n$ particles with momentum $\vec{q}$. Consider in separate the cases in which $\vec{q}\neq\pm\vec{k}$ and the case in which $\vec{q}=\vec{k}$.